9 36: Difference between revisions

Revision as of 20:13, 28 August 2005

9_35

9_37

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Visit 9 36's page at the original Knot Atlas!

9 36 Quick Notes

9 36 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_7,10,8,11 X₃₉₄₈ X_9,3,10,2 X_11,17,12,16 X_5,15,6,14 X_15,7,16,6 X_13,1,14,18 X_17,13,18,12
Gauss code	-1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -5, 9, -8, 6, -7, 5, -9, 8
Dowker-Thistlethwaite code	4 8 14 10 2 16 18 6 12
Conway Notation	[22,3,2]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	2
3-genus	3
Bridge index	3
Super bridge index	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,7\}}
Nakanishi index	1
Maximal Thurston-Bennequin number	[1][-12]
Hyperbolic Volume	9.88458
A-Polynomial	See Data:9 36/A-polynomial

[edit Notes for 9 36's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2}
Topological 4 genus	$2$
Concordance genus	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3}
Rasmussen s-Invariant	4

[edit Notes for 9 36's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-8t+9-8t^{-1}+5t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 37, 4 }
Jones polynomial	$-q^{9}+2q^{8}-4q^{7}+6q^{6}-6q^{5}+6q^{4}-5q^{3}+4q^{2}-2q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}+z^{4}a^{-2}-4z^{4}a^{-4}+2z^{4}a^{-6}+3z^{2}a^{-2}-5z^{2}a^{-4}+6z^{2}a^{-6}-z^{2}a^{-8}+2a^{-2}-3a^{-4}+4a^{-6}-2a^{-8}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-4}+z^{8}a^{-6}+2z^{7}a^{-3}+5z^{7}a^{-5}+3z^{7}a^{-7}+z^{6}a^{-2}+z^{6}a^{-4}+4z^{6}a^{-6}+4z^{6}a^{-8}-7z^{5}a^{-3}-14z^{5}a^{-5}-4z^{5}a^{-7}+3z^{5}a^{-9}-4z^{4}a^{-2}-12z^{4}a^{-4}-17z^{4}a^{-6}-7z^{4}a^{-8}+2z^{4}a^{-10}+6z^{3}a^{-3}+9z^{3}a^{-5}-2z^{3}a^{-9}+z^{3}a^{-11}+5z^{2}a^{-2}+12z^{2}a^{-4}+15z^{2}a^{-6}+7z^{2}a^{-8}-z^{2}a^{-10}-za^{-3}-2za^{-5}+za^{-7}+za^{-9}-za^{-11}-2a^{-2}-3a^{-4}-4a^{-6}-2a^{-8}$
The A2 invariant	$1+q^{-4}+q^{-6}-q^{-8}+q^{-10}-2q^{-12}+q^{-14}+q^{-16}+q^{-18}+2q^{-20}-q^{-22}-q^{-26}-q^{-28}$
The G2 invariant	$q^{-2}-q^{-4}+4q^{-6}-5q^{-8}+5q^{-10}-2q^{-12}-4q^{-14}+14q^{-16}-17q^{-18}+19q^{-20}-11q^{-22}-2q^{-24}+18q^{-26}-27q^{-28}+28q^{-30}-17q^{-32}+q^{-34}+13q^{-36}-23q^{-38}+20q^{-40}-9q^{-42}-5q^{-44}+15q^{-46}-18q^{-48}+9q^{-50}+3q^{-52}-18q^{-54}+24q^{-56}-24q^{-58}+16q^{-60}+q^{-62}-18q^{-64}+32q^{-66}-33q^{-68}+28q^{-70}-9q^{-72}-10q^{-74}+25q^{-76}-28q^{-78}+24q^{-80}-7q^{-82}-7q^{-84}+19q^{-86}-16q^{-88}+6q^{-90}+6q^{-92}-16q^{-94}+18q^{-96}-11q^{-98}-2q^{-100}+11q^{-102}-18q^{-104}+20q^{-106}-15q^{-108}+4q^{-110}+3q^{-112}-12q^{-114}+12q^{-116}-13q^{-118}+10q^{-120}-5q^{-122}+q^{-124}+3q^{-126}-7q^{-128}+6q^{-130}-5q^{-132}+4q^{-134}-2q^{-136}+q^{-140}-2q^{-142}+2q^{-144}-q^{-146}+q^{-148}$

Further Quantum Invariants

Further quantum knot invariants for 9_36.

A1 Invariants.

Weight	Invariant
1	$q-q^{-1}+2q^{-3}-q^{-5}+q^{-7}+2q^{-13}-2q^{-15}+q^{-17}-q^{-19}$
2	$q^{6}-q^{4}-2q^{2}+4+q^{-2}-6q^{-4}+4q^{-6}+5q^{-8}-6q^{-10}+q^{-12}+7q^{-14}-4q^{-16}-3q^{-18}+5q^{-20}-4q^{-24}+q^{-26}+4q^{-28}-3q^{-30}-5q^{-32}+7q^{-34}-7q^{-38}+5q^{-40}+q^{-42}-4q^{-44}+2q^{-46}-q^{-50}+q^{-52}$
3	$q^{15}-q^{13}-2q^{11}+5q^{7}+3q^{5}-7q^{3}-8q+6q^{-1}+14q^{-3}-18q^{-7}-6q^{-9}+17q^{-11}+16q^{-13}-11q^{-15}-21q^{-17}+5q^{-19}+23q^{-21}+4q^{-23}-21q^{-25}-10q^{-27}+20q^{-29}+15q^{-31}-16q^{-33}-17q^{-35}+12q^{-37}+19q^{-39}-9q^{-41}-21q^{-43}+3q^{-45}+18q^{-47}+3q^{-49}-16q^{-51}-12q^{-53}+10q^{-55}+20q^{-57}-2q^{-59}-24q^{-61}-5q^{-63}+22q^{-65}+14q^{-67}-20q^{-69}-14q^{-71}+12q^{-73}+12q^{-75}-6q^{-77}-8q^{-79}+3q^{-81}+4q^{-83}-2q^{-85}-q^{-87}+q^{-89}+q^{-97}-q^{-99}$
4	$q^{28}-q^{26}-2q^{24}+q^{20}+7q^{18}+q^{16}-7q^{14}-9q^{12}-8q^{10}+17q^{8}+19q^{6}+5q^{4}-17q^{2}-39-2q^{-2}+28q^{-4}+44q^{-6}+18q^{-8}-51q^{-10}-49q^{-12}-17q^{-14}+51q^{-16}+78q^{-18}+3q^{-20}-54q^{-22}-83q^{-24}-10q^{-26}+86q^{-28}+73q^{-30}+8q^{-32}-94q^{-34}-82q^{-36}+29q^{-38}+93q^{-40}+71q^{-42}-55q^{-44}-104q^{-46}-27q^{-48}+70q^{-50}+92q^{-52}-15q^{-54}-91q^{-56}-48q^{-58}+52q^{-60}+85q^{-62}-77q^{-66}-54q^{-68}+37q^{-70}+75q^{-72}+23q^{-74}-56q^{-76}-71q^{-78}-q^{-80}+55q^{-82}+69q^{-84}+4q^{-86}-75q^{-88}-69q^{-90}-6q^{-92}+99q^{-94}+89q^{-96}-29q^{-98}-105q^{-100}-89q^{-102}+60q^{-104}+126q^{-106}+38q^{-108}-69q^{-110}-113q^{-112}-2q^{-114}+81q^{-116}+55q^{-118}-8q^{-120}-67q^{-122}-22q^{-124}+25q^{-126}+26q^{-128}+11q^{-130}-22q^{-132}-8q^{-134}+4q^{-136}+3q^{-138}+7q^{-140}-6q^{-142}-q^{-144}+q^{-146}-q^{-148}+3q^{-150}-2q^{-152}-q^{-158}+q^{-160}$
5	$q^{45}-q^{43}-2q^{41}+q^{37}+3q^{35}+5q^{33}+q^{31}-9q^{29}-11q^{27}-6q^{25}+5q^{23}+21q^{21}+25q^{19}+6q^{17}-27q^{15}-44q^{13}-34q^{11}+8q^{9}+59q^{7}+76q^{5}+35q^{3}-45q-106q^{-1}-100q^{-3}-15q^{-5}+100q^{-7}+164q^{-9}+110q^{-11}-40q^{-13}-176q^{-15}-207q^{-17}-84q^{-19}+125q^{-21}+269q^{-23}+221q^{-25}+2q^{-27}-242q^{-29}-333q^{-31}-173q^{-33}+140q^{-35}+375q^{-37}+336q^{-39}+30q^{-41}-322q^{-43}-450q^{-45}-221q^{-47}+200q^{-49}+482q^{-51}+381q^{-53}-35q^{-55}-433q^{-57}-485q^{-59}-128q^{-61}+338q^{-63}+519q^{-65}+253q^{-67}-222q^{-69}-494q^{-71}-330q^{-73}+118q^{-75}+436q^{-77}+348q^{-79}-47q^{-81}-372q^{-83}-329q^{-85}+8q^{-87}+313q^{-89}+296q^{-91}-3q^{-93}-278q^{-95}-265q^{-97}+8q^{-99}+260q^{-101}+253q^{-103}+5q^{-105}-247q^{-107}-268q^{-109}-46q^{-111}+216q^{-113}+294q^{-115}+133q^{-117}-138q^{-119}-318q^{-121}-258q^{-123}+15q^{-125}+301q^{-127}+383q^{-129}+166q^{-131}-219q^{-133}-481q^{-135}-364q^{-137}+81q^{-139}+496q^{-141}+531q^{-143}+115q^{-145}-430q^{-147}-633q^{-149}-289q^{-151}+295q^{-153}+616q^{-155}+423q^{-157}-124q^{-159}-530q^{-161}-459q^{-163}-22q^{-165}+380q^{-167}+414q^{-169}+112q^{-171}-228q^{-173}-320q^{-175}-139q^{-177}+112q^{-179}+213q^{-181}+116q^{-183}-39q^{-185}-118q^{-187}-82q^{-189}+6q^{-191}+60q^{-193}+48q^{-195}+q^{-197}-25q^{-199}-22q^{-201}-4q^{-203}+9q^{-205}+11q^{-207}+2q^{-209}-7q^{-211}-3q^{-213}+2q^{-215}+2q^{-219}+q^{-221}-3q^{-223}-q^{-225}+2q^{-227}+q^{-233}-q^{-235}$

A2 Invariants.

Weight	Invariant
1,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1+ q^{-4} + q^{-6} - q^{-8} + q^{-10} -2 q^{-12} + q^{-14} + q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-26} - q^{-28} }
1,1	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-2 q^2+8-16 q^{-2} +29 q^{-4} -44 q^{-6} +60 q^{-8} -74 q^{-10} +80 q^{-12} -72 q^{-14} +62 q^{-16} -32 q^{-18} +3 q^{-20} +38 q^{-22} -70 q^{-24} +100 q^{-26} -123 q^{-28} +132 q^{-30} -134 q^{-32} +118 q^{-34} -96 q^{-36} +64 q^{-38} -30 q^{-40} +28 q^{-44} -44 q^{-46} +52 q^{-48} -52 q^{-50} +46 q^{-52} -44 q^{-54} +34 q^{-56} -28 q^{-58} +24 q^{-60} -20 q^{-62} +16 q^{-64} -12 q^{-66} +9 q^{-68} -6 q^{-70} +4 q^{-72} -2 q^{-74} + q^{-76} }
2,0	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^4-1+2 q^{-4} + q^{-6} -2 q^{-8} - q^{-10} +4 q^{-12} + q^{-14} -2 q^{-16} +2 q^{-18} +3 q^{-20} - q^{-24} +2 q^{-26} +2 q^{-28} - q^{-30} +2 q^{-32} + q^{-34} -4 q^{-36} -3 q^{-38} +2 q^{-40} -2 q^{-42} -3 q^{-44} +2 q^{-46} +4 q^{-48} + q^{-50} -2 q^{-52} + q^{-54} -3 q^{-58} -2 q^{-60} - q^{-62} + q^{-66} + q^{-68} + q^{-70} }

A3 Invariants.

Weight

Invariant

0,1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1- q^{-2} +2 q^{-4} +2 q^{-6} -2 q^{-8} +4 q^{-10} + q^{-12} -4 q^{-14} +4 q^{-16} - q^{-18} -5 q^{-20} +4 q^{-22} +2 q^{-24} -3 q^{-26} +3 q^{-28} +3 q^{-30} + q^{-32} - q^{-34} + q^{-36} +3 q^{-38} -5 q^{-40} -2 q^{-42} +4 q^{-44} -6 q^{-46} -2 q^{-48} +5 q^{-50} -3 q^{-52} -2 q^{-54} +3 q^{-56} - q^{-60} + q^{-62} }

1,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-1} +2 q^{-5} +2 q^{-9} - q^{-11} -2 q^{-15} - q^{-17} + q^{-21} +3 q^{-23} + q^{-25} +3 q^{-27} - q^{-29} -2 q^{-33} - q^{-35} - q^{-37} }

A4 Invariants.

Weight

Invariant

0,1,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} + q^{-6} +3 q^{-8} + q^{-10} + q^{-12} +3 q^{-14} -3 q^{-18} + q^{-22} -3 q^{-24} - q^{-26} +6 q^{-28} +5 q^{-30} -4 q^{-32} +2 q^{-34} +5 q^{-36} -5 q^{-38} -5 q^{-40} +4 q^{-42} + q^{-44} - q^{-46} +5 q^{-48} +6 q^{-50} - q^{-52} -2 q^{-54} +2 q^{-56} -3 q^{-58} -8 q^{-60} -2 q^{-62} + q^{-64} -4 q^{-66} -2 q^{-68} +2 q^{-70} +2 q^{-72} + q^{-78} + q^{-80} }

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} +2 q^{-6} + q^{-8} + q^{-10} +2 q^{-12} - q^{-14} -3 q^{-18} - q^{-20} -2 q^{-22} + q^{-26} +3 q^{-28} +3 q^{-30} +2 q^{-32} +3 q^{-34} - q^{-36} -2 q^{-40} -2 q^{-42} - q^{-44} - q^{-46} }

B2 Invariants.

Weight

Invariant

0,1

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1- q^{-2} +4 q^{-4} -4 q^{-6} +6 q^{-8} -6 q^{-10} +7 q^{-12} -6 q^{-14} +4 q^{-16} -3 q^{-18} - q^{-20} +4 q^{-22} -8 q^{-24} +11 q^{-26} -11 q^{-28} +13 q^{-30} -11 q^{-32} +11 q^{-34} -7 q^{-36} +5 q^{-38} - q^{-40} -2 q^{-42} +4 q^{-44} -6 q^{-46} +6 q^{-48} -7 q^{-50} +5 q^{-52} -4 q^{-54} +3 q^{-56} -2 q^{-58} + q^{-60} - q^{-62} }

1,0

q^{2}-q^{-2}-q^{-4}+3q^{-6}+3q^{-8}-2q^{-10}-4q^{-12}+q^{-14}+6q^{-16}+3q^{-18}-6q^{-20}-5q^{-22}+4q^{-24}+7q^{-26}-7q^{-30}-3q^{-32}+5q^{-34}+5q^{-36}-2q^{-38}-4q^{-40}+2q^{-42}+4q^{-44}-4q^{-48}+5q^{-52}+2q^{-54}-5q^{-56}-3q^{-58}+4q^{-60}+5q^{-62}-3q^{-64}-6q^{-66}+6q^{-70}+q^{-72}-6q^{-74}-5q^{-76}+2q^{-78}+6q^{-80}-4q^{-84}-3q^{-86}+q^{-88}+3q^{-90}+q^{-92}-q^{-94}-q^{-96}+q^{-100}

D4 Invariants.

Weight

Invariant

1,0,0,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} - q^{-4} +3 q^{-6} -2 q^{-8} +6 q^{-10} -3 q^{-12} +6 q^{-14} -4 q^{-16} +7 q^{-18} -5 q^{-20} +2 q^{-22} -4 q^{-24} - q^{-28} -4 q^{-30} +4 q^{-32} -6 q^{-34} +9 q^{-36} -7 q^{-38} +12 q^{-40} -6 q^{-42} +12 q^{-44} -7 q^{-46} +9 q^{-48} -5 q^{-50} +4 q^{-52} -4 q^{-54} -2 q^{-56} - q^{-58} -4 q^{-60} +3 q^{-62} -6 q^{-64} +3 q^{-66} -5 q^{-68} +6 q^{-70} -4 q^{-72} +2 q^{-74} -3 q^{-76} +3 q^{-78} - q^{-80} + q^{-82} - q^{-84} + q^{-86} }

G2 Invariants.

Weight

Invariant

1,0

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{-2} - q^{-4} +4 q^{-6} -5 q^{-8} +5 q^{-10} -2 q^{-12} -4 q^{-14} +14 q^{-16} -17 q^{-18} +19 q^{-20} -11 q^{-22} -2 q^{-24} +18 q^{-26} -27 q^{-28} +28 q^{-30} -17 q^{-32} + q^{-34} +13 q^{-36} -23 q^{-38} +20 q^{-40} -9 q^{-42} -5 q^{-44} +15 q^{-46} -18 q^{-48} +9 q^{-50} +3 q^{-52} -18 q^{-54} +24 q^{-56} -24 q^{-58} +16 q^{-60} + q^{-62} -18 q^{-64} +32 q^{-66} -33 q^{-68} +28 q^{-70} -9 q^{-72} -10 q^{-74} +25 q^{-76} -28 q^{-78} +24 q^{-80} -7 q^{-82} -7 q^{-84} +19 q^{-86} -16 q^{-88} +6 q^{-90} +6 q^{-92} -16 q^{-94} +18 q^{-96} -11 q^{-98} -2 q^{-100} +11 q^{-102} -18 q^{-104} +20 q^{-106} -15 q^{-108} +4 q^{-110} +3 q^{-112} -12 q^{-114} +12 q^{-116} -13 q^{-118} +10 q^{-120} -5 q^{-122} + q^{-124} +3 q^{-126} -7 q^{-128} +6 q^{-130} -5 q^{-132} +4 q^{-134} -2 q^{-136} + q^{-140} -2 q^{-142} +2 q^{-144} - q^{-146} + q^{-148} }

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["9 36"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= $-t^{3}+5t^{2}-8t+9-8t^{-1}+5t^{-2}-t^{-3}$

In[5]:= Conway[K][z]

Out[5]= $-z^{6}-z^{4}+3z^{2}+1$

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= $\{1\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 37, 4 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= $-q^{9}+2q^{8}-4q^{7}+6q^{6}-6q^{5}+6q^{4}-5q^{3}+4q^{2}-2q+1$

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= $-z^{6}a^{-4}+z^{4}a^{-2}-4z^{4}a^{-4}+2z^{4}a^{-6}+3z^{2}a^{-2}-5z^{2}a^{-4}+6z^{2}a^{-6}-z^{2}a^{-8}+2a^{-2}-3a^{-4}+4a^{-6}-2a^{-8}$

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= $z^{8}a^{-4}+z^{8}a^{-6}+2z^{7}a^{-3}+5z^{7}a^{-5}+3z^{7}a^{-7}+z^{6}a^{-2}+z^{6}a^{-4}+4z^{6}a^{-6}+4z^{6}a^{-8}-7z^{5}a^{-3}-14z^{5}a^{-5}-4z^{5}a^{-7}+3z^{5}a^{-9}-4z^{4}a^{-2}-12z^{4}a^{-4}-17z^{4}a^{-6}-7z^{4}a^{-8}+2z^{4}a^{-10}+6z^{3}a^{-3}+9z^{3}a^{-5}-2z^{3}a^{-9}+z^{3}a^{-11}+5z^{2}a^{-2}+12z^{2}a^{-4}+15z^{2}a^{-6}+7z^{2}a^{-8}-z^{2}a^{-10}-za^{-3}-2za^{-5}+za^{-7}+za^{-9}-za^{-11}-2a^{-2}-3a^{-4}-4a^{-6}-2a^{-8}$

Vassiliev invariants

V₂ and V₃:

(3, 7)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 12}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 56}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 72}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 270}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 42}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 672}

{\frac {4304}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{704}{3}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 216}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 288}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1568}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3240}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 504}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{78751}{10}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{138}{5}}

{\frac {48742}{15}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{577}{6}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4191}{10}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 4 is the signature of 9 36. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-2

-1

0

1

2

3

4

5

6

7

χ

19

1

-1

17

1

15

3

1

-2

13

3

1

2

11

3

0

9

3

0

7

2

3

1

5

2

3

-1

3

1

3

2

1

-1

1

Integral Khovanov Homology

(db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=5}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1}	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{3}}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=6}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=7}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{Include}(\textrm{ColouredJonesM.mhtml})}

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[9, 36]]
Out[2]=	9
In[3]:=	PD[Knot[9, 36]]
Out[3]=	PD[X[1, 4, 2, 5], X[7, 10, 8, 11], X[3, 9, 4, 8], X[9, 3, 10, 2], X[11, 17, 12, 16], X[5, 15, 6, 14], X[15, 7, 16, 6], X[13, 1, 14, 18], X[17, 13, 18, 12]]
In[4]:=	GaussCode[Knot[9, 36]]
Out[4]=	GaussCode[-1, 4, -3, 1, -6, 7, -2, 3, -4, 2, -5, 9, -8, 6, -7, 5, -9, 8]
In[5]:=	BR[Knot[9, 36]]
Out[5]=	BR[4, {1, 1, 1, -2, 1, 1, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[9, 36]][t]
Out[6]=	-3 5 8 2 3 9 - t + -- - - - 8 t + 5 t - t 2 t t
In[7]:=	Conway[Knot[9, 36]][z]
Out[7]=	2 4 6 1 + 3 z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 36]}
In[9]:=	{KnotDet[Knot[9, 36]], KnotSignature[Knot[9, 36]]}
Out[9]=	{37, 4}
In[10]:=	J=Jones[Knot[9, 36]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 2 q + 4 q - 5 q + 6 q - 6 q + 6 q - 4 q + 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 36], Knot[11, NonAlternating, 16]}
In[12]:=	A2Invariant[Knot[9, 36]][q]
Out[12]=	4 6 8 10 12 14 16 18 20 22 26 1 + q + q - q + q - 2 q + q + q + q + 2 q - q - q - 28 q
In[13]:=	Kauffman[Knot[9, 36]][a, z]
Out[13]=	2 2 2 -2 4 3 2 z z z 2 z z z 7 z 15 z -- - -- - -- - -- - --- + -- + -- - --- - -- - --- + ---- + ----- + 8 6 4 2 11 9 7 5 3 10 8 6 a a a a a a a a a a a a 2 2 3 3 3 3 4 4 4 12 z 5 z z 2 z 9 z 6 z 2 z 7 z 17 z ----- + ---- + --- - ---- + ---- + ---- + ---- - ---- - ----- - 4 2 11 9 5 3 10 8 6 a a a a a a a a a 4 4 5 5 5 5 6 6 6 6 12 z 4 z 3 z 4 z 14 z 7 z 4 z 4 z z z ----- - ---- + ---- - ---- - ----- - ---- + ---- + ---- + -- + -- + 4 2 9 7 5 3 8 6 4 2 a a a a a a a a a a 7 7 7 8 8 3 z 5 z 2 z z z ---- + ---- + ---- + -- + -- 7 5 3 6 4 a a a a a
In[14]:=	{Vassiliev[2][Knot[9, 36]], Vassiliev[3][Knot[9, 36]]}
Out[14]=	{0, 7}
In[15]:=	Kh[Knot[9, 36]][q, t]
Out[15]=	3 3 5 1 q q 5 7 7 2 9 2 3 q + 2 q + ---- + - + -- + 3 q t + 2 q t + 3 q t + 3 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 3 q t + 3 q t + 3 q t + 3 q t + q t + 3 q t + 15 6 17 6 19 7 q t + q t + q t

@@ Line 1: / Line 1: @@
 <!--  -->
+<!-- -->
+<!--  -->
+<!-- -->
 <!-- provide an anchor so we can return to the top of the page -->
 <span id="top"></span>
+<!-- -->
 <!-- this relies on transclusion for next and previous links -->
 {{Knot Navigation Links|ext=gif}}
+{{Rolfsen Knot Page Header|n=9|k=36|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-3,1,-6,7,-2,3,-4,2,-5,9,-8,6,-7,5,-9,8/goTop.html}}
-{| align=left
-|- valign=top
-|[[Image:{{PAGENAME}}.gif]]
-|{{Rolfsen Knot Site Links|n=9|k=36|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/-1,4,-3,1,-6,7,-2,3,-4,2,-5,9,-8,6,-7,5,-9,8/goTop.html}}
-|{{:{{PAGENAME}} Quick Notes}}
-|}
 <br style="clear:both" />
@@ Line 24: / Line 21: @@
 {{Vassiliev Invariants}}
-===[[Khovanov Homology]]===
+{{Khovanov Homology|table=<table border=1>
-The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
-<center><table border=1>
 <tr align=center>
 <td width=14.2857%><table cellpadding=0 cellspacing=0>
@@ Line 47: / Line 40: @@
 <tr align=center><td>1</td><td bgcolor=yellow>&nbsp;</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>-1</td></tr>
 <tr align=center><td>-1</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
-</table></center>
+</table>}}
 {{Computer Talk Header}}
@@ Line 132: / Line 124: @@
   q   t  + q   t  + q   t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

9 36: Difference between revisions

Revision as of 20:13, 28 August 2005

Contents

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Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

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