9 11

9_10

9_12

Visit 9 11's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 11's page at Knotilus! Visit 9 11's page at the original Knot Atlas! 9 11 Quick Notes

9 11 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X13,1,14,18 X5,15,6,14 X7,17,8,16 X15,7,16,6 X17,9,18,8
Gauss code	-1, 4, -3, 1, -6, 8, -7, 9, -2, 3, -4, 2, -5, 6, -8, 7, -9, 5
Dowker-Thistlethwaite code	4 10 14 16 12 2 18 6 8
Conway Notation	[4122]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 2 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 \{4,6\}} Nakanishi index 1 Maximal Thurston-Bennequin number [1][-12] Hyperbolic Volume 8.28859 A-Polynomial See Data:9 11/A-polynomial

[edit Notes for 9 11'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 11's four dimensional invariants]

Polynomial invariants

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

Further Quantum Invariants

Further quantum knot invariants for 9_11.

A1 Invariants.

Weight	Invariant
1	$q-q^{-1}+q^{-3}-q^{-5}+2q^{-7}+q^{-9}+q^{-13}-2q^{-15}+q^{-17}-q^{-19}$
2	$q^{6}-q^{4}-2q^{2}+3+q^{-2}-4q^{-4}+3q^{-6}+4q^{-8}-4q^{-10}+q^{-12}+4q^{-14}-4q^{-16}-q^{-18}+4q^{-20}-2q^{-24}+2q^{-26}+3q^{-28}-2q^{-30}-4q^{-32}+4q^{-34}-q^{-36}-5q^{-38}+4q^{-40}-q^{-42}-3q^{-44}+3q^{-46}-q^{-50}+q^{-52}$
3	$q^{15}-q^{13}-2q^{11}+4q^{7}+3q^{5}-5q^{3}-6q+4q^{-1}+9q^{-3}-11q^{-7}-4q^{-9}+11q^{-11}+8q^{-13}-7q^{-15}-10q^{-17}+4q^{-19}+13q^{-21}+q^{-23}-9q^{-25}-3q^{-27}+10q^{-29}+4q^{-31}-8q^{-33}-8q^{-35}+8q^{-37}+8q^{-39}-7q^{-41}-9q^{-43}+4q^{-45}+10q^{-47}-8q^{-51}-6q^{-53}+6q^{-55}+8q^{-57}-q^{-59}-14q^{-61}-2q^{-63}+10q^{-65}+5q^{-67}-9q^{-69}-4q^{-71}+6q^{-73}+2q^{-75}-3q^{-77}+3q^{-81}-q^{-83}-2q^{-85}+2q^{-87}+2q^{-89}-q^{-91}-q^{-93}+q^{-97}-q^{-99}$
4	$q^{28}-q^{26}-2q^{24}+q^{20}+6q^{18}+q^{16}-5q^{14}-7q^{12}-7q^{10}+12q^{8}+13q^{6}+3q^{4}-10q^{2}-24+16q^{-4}+23q^{-6}+9q^{-8}-29q^{-10}-21q^{-12}-4q^{-14}+25q^{-16}+33q^{-18}-5q^{-20}-21q^{-22}-31q^{-24}+35q^{-28}+21q^{-30}+3q^{-32}-36q^{-34}-28q^{-36}+14q^{-38}+30q^{-40}+30q^{-42}-20q^{-44}-37q^{-46}-6q^{-48}+23q^{-50}+36q^{-52}-9q^{-54}-36q^{-56}-13q^{-58}+20q^{-60}+30q^{-62}-7q^{-64}-32q^{-66}-16q^{-68}+19q^{-70}+29q^{-72}+3q^{-74}-25q^{-76}-27q^{-78}+6q^{-80}+22q^{-82}+24q^{-84}+q^{-86}-29q^{-88}-25q^{-90}-6q^{-92}+38q^{-94}+35q^{-96}-11q^{-98}-37q^{-100}-38q^{-102}+22q^{-104}+47q^{-106}+14q^{-108}-18q^{-110}-45q^{-112}-q^{-114}+28q^{-116}+17q^{-118}+5q^{-120}-26q^{-122}-7q^{-124}+7q^{-126}+5q^{-128}+11q^{-130}-8q^{-132}-2q^{-134}-2q^{-136}-3q^{-138}+7q^{-140}-2q^{-142}+q^{-144}-q^{-146}-3q^{-148}+2q^{-150}-q^{-152}+q^{-154}-q^{-158}+q^{-160}$
5	$q^{45}-q^{43}-2q^{41}+q^{37}+3q^{35}+4q^{33}+q^{31}-7q^{29}-9q^{27}-5q^{25}+3q^{23}+15q^{21}+18q^{19}+5q^{17}-18q^{15}-28q^{13}-20q^{11}+5q^{9}+35q^{7}+41q^{5}+16q^{3}-27q-53q^{-1}-43q^{-3}-q^{-5}+51q^{-7}+67q^{-9}+35q^{-11}-27q^{-13}-69q^{-15}-66q^{-17}-16q^{-19}+51q^{-21}+84q^{-23}+59q^{-25}-9q^{-27}-73q^{-29}-89q^{-31}-43q^{-33}+41q^{-35}+100q^{-37}+92q^{-39}+10q^{-41}-83q^{-43}-118q^{-45}-67q^{-47}+48q^{-49}+132q^{-51}+111q^{-53}-7q^{-55}-120q^{-57}-139q^{-59}-44q^{-61}+98q^{-63}+156q^{-65}+73q^{-67}-67q^{-69}-145q^{-71}-97q^{-73}+40q^{-75}+135q^{-77}+98q^{-79}-23q^{-81}-112q^{-83}-94q^{-85}+12q^{-87}+99q^{-89}+81q^{-91}-16q^{-93}-90q^{-95}-73q^{-97}+16q^{-99}+86q^{-101}+69q^{-103}-18q^{-105}-88q^{-107}-75q^{-109}+4q^{-111}+75q^{-113}+87q^{-115}+31q^{-117}-52q^{-119}-90q^{-121}-74q^{-123}+5q^{-125}+91q^{-127}+114q^{-129}+55q^{-131}-56q^{-133}-148q^{-135}-122q^{-137}+18q^{-139}+149q^{-141}+166q^{-143}+51q^{-145}-127q^{-147}-199q^{-149}-96q^{-151}+86q^{-153}+190q^{-155}+132q^{-157}-37q^{-159}-167q^{-161}-141q^{-163}-q^{-165}+127q^{-167}+128q^{-169}+23q^{-171}-82q^{-173}-104q^{-175}-34q^{-177}+51q^{-179}+74q^{-181}+33q^{-183}-25q^{-185}-47q^{-187}-29q^{-189}+8q^{-191}+28q^{-193}+22q^{-195}-q^{-197}-15q^{-199}-14q^{-201}-4q^{-203}+6q^{-205}+9q^{-207}+4q^{-209}-4q^{-211}-3q^{-213}-q^{-215}-q^{-217}+2q^{-219}+2q^{-221}-q^{-223}+q^{-227}-q^{-229}+q^{-233}-q^{-235}$

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^{-8} +2 q^{-10} +2 q^{-14} + q^{-16} + 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+6-12 q^{-2} +19 q^{-4} -30 q^{-6} +40 q^{-8} -44 q^{-10} +49 q^{-12} -44 q^{-14} +38 q^{-16} -18 q^{-18} +3 q^{-20} +18 q^{-22} -34 q^{-24} +54 q^{-26} -68 q^{-28} +74 q^{-30} -76 q^{-32} +74 q^{-34} -63 q^{-36} +50 q^{-38} -32 q^{-40} +14 q^{-42} -2 q^{-44} -10 q^{-46} +14 q^{-48} -22 q^{-50} +25 q^{-52} -26 q^{-54} +24 q^{-56} -28 q^{-58} +26 q^{-60} -22 q^{-62} +18 q^{-64} -14 q^{-66} +11 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- q^{-2} + q^{-4} + q^{-6} -2 q^{-8} +4 q^{-12} +3 q^{-14} - q^{-16} + q^{-18} +2 q^{-20} - q^{-22} - q^{-24} + q^{-28} -2 q^{-30} +3 q^{-32} +3 q^{-34} + q^{-38} +4 q^{-40} -4 q^{-44} - q^{-46} -2 q^{-50} -4 q^{-52} - q^{-54} - q^{-58} + 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} + q^{-4} + q^{-6} -2 q^{-8} +3 q^{-10} + q^{-12} -2 q^{-14} +3 q^{-16} -4 q^{-20} +3 q^{-22} +3 q^{-24} -2 q^{-26} +5 q^{-28} +4 q^{-30} +2 q^{-32} - q^{-34} -6 q^{-40} -3 q^{-42} +2 q^{-44} -4 q^{-46} -2 q^{-48} +5 q^{-50} -2 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} + q^{-5} - q^{-7} + q^{-9} - q^{-11} + q^{-13} + q^{-17} + q^{-19} + q^{-21} +2 q^{-23} +2 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^{-8} + q^{-14} + q^{-16} +2 q^{-20} +2 q^{-22} - q^{-24} - q^{-26} +2 q^{-28} +2 q^{-30} -4 q^{-32} +2 q^{-34} +6 q^{-36} +2 q^{-38} + q^{-40} +7 q^{-42} +4 q^{-44} + q^{-48} -5 q^{-52} -6 q^{-54} -2 q^{-56} -4 q^{-58} -6 q^{-60} - q^{-62} +2 q^{-64} -2 q^{-66} - 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} + q^{-6} + q^{-12} - q^{-14} + q^{-16} - q^{-18} + q^{-20} + q^{-24} + q^{-26} +2 q^{-28} +2 q^{-30} + q^{-32} +2 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} +3 q^{-4} -3 q^{-6} +4 q^{-8} -5 q^{-10} +5 q^{-12} -4 q^{-14} +3 q^{-16} -2 q^{-18} +3 q^{-22} -5 q^{-24} +8 q^{-26} -7 q^{-28} +10 q^{-30} -8 q^{-32} +9 q^{-34} -6 q^{-36} +4 q^{-38} -2 q^{-40} - q^{-42} +2 q^{-44} -4 q^{-46} +4 q^{-48} -5 q^{-50} +4 q^{-52} -4 q^{-54} +3 q^{-56} -2 q^{-58} + q^{-60} - q^{-62} }
1,0	$q^{2}-q^{-2}-q^{-4}+2q^{-6}+2q^{-8}-2q^{-10}-3q^{-12}+q^{-14}+5q^{-16}+2q^{-18}-4q^{-20}-3q^{-22}+3q^{-24}+5q^{-26}-5q^{-30}-3q^{-32}+3q^{-34}+4q^{-36}-3q^{-40}+2q^{-42}+4q^{-44}+2q^{-46}-2q^{-48}+q^{-50}+4q^{-52}+2q^{-54}-4q^{-56}-3q^{-58}+2q^{-60}+2q^{-62}-3q^{-64}-5q^{-66}-q^{-68}+3q^{-70}+q^{-72}-4q^{-74}-4q^{-76}+q^{-78}+5q^{-80}+q^{-82}-3q^{-84}-3q^{-86}+q^{-88}+3q^{-90}+q^{-92}-q^{-94}-q^{-96}+q^{-100}$

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} +2 q^{-6} -2 q^{-8} +4 q^{-10} -3 q^{-12} +4 q^{-14} -3 q^{-16} +5 q^{-18} -3 q^{-20} +2 q^{-22} -2 q^{-24} + q^{-26} -2 q^{-30} +3 q^{-32} -3 q^{-34} +7 q^{-36} -4 q^{-38} +10 q^{-40} -3 q^{-42} +10 q^{-44} -5 q^{-46} +7 q^{-48} -5 q^{-50} +2 q^{-52} -5 q^{-54} -3 q^{-56} -3 q^{-58} -3 q^{-60} + q^{-62} -4 q^{-64} +2 q^{-66} -3 q^{-68} +5 q^{-70} -3 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} +3 q^{-6} -4 q^{-8} +3 q^{-10} - q^{-12} -2 q^{-14} +10 q^{-16} -12 q^{-18} +13 q^{-20} -7 q^{-22} -2 q^{-24} +10 q^{-26} -17 q^{-28} +17 q^{-30} -12 q^{-32} +2 q^{-34} +8 q^{-36} -13 q^{-38} +12 q^{-40} -7 q^{-42} -2 q^{-44} +7 q^{-46} -8 q^{-48} +4 q^{-50} - q^{-52} -5 q^{-54} +16 q^{-56} -12 q^{-58} +10 q^{-60} - q^{-62} -8 q^{-64} +19 q^{-66} -20 q^{-68} +18 q^{-70} -9 q^{-72} +16 q^{-76} -19 q^{-78} +17 q^{-80} -8 q^{-82} -2 q^{-84} +8 q^{-86} -10 q^{-88} +4 q^{-90} -4 q^{-94} +7 q^{-96} -6 q^{-98} +3 q^{-102} -10 q^{-104} +10 q^{-106} -9 q^{-108} +4 q^{-110} - q^{-112} -4 q^{-114} +8 q^{-116} -11 q^{-118} +11 q^{-120} -7 q^{-122} +2 q^{-124} + q^{-126} -6 q^{-128} +6 q^{-130} -6 q^{-132} +5 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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 11"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+5t^{2}-7t+7-7t^{-1}+5t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}-z^{4}+4z^{2}+1$

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]=

{ 33, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{6}a^{-4}+z^{4}a^{-2}-4z^{4}a^{-4}+2z^{4}a^{-6}+3z^{2}a^{-2}-4z^{2}a^{-4}+6z^{2}a^{-6}-z^{2}a^{-8}+a^{-2}-a^{-4}+3a^{-6}-2a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{8}a^{-4}+z^{8}a^{-6}+2z^{7}a^{-3}+4z^{7}a^{-5}+2z^{7}a^{-7}+z^{6}a^{-2}-z^{6}a^{-4}+z^{6}a^{-6}+3z^{6}a^{-8}-8z^{5}a^{-3}-12z^{5}a^{-5}-z^{5}a^{-7}+3z^{5}a^{-9}-4z^{4}a^{-2}-5z^{4}a^{-4}-7z^{4}a^{-6}-4z^{4}a^{-8}+2z^{4}a^{-10}+8z^{3}a^{-3}+9z^{3}a^{-5}-3z^{3}a^{-7}-3z^{3}a^{-9}+z^{3}a^{-11}+4z^{2}a^{-2}+5z^{2}a^{-4}+6z^{2}a^{-6}+4z^{2}a^{-8}-z^{2}a^{-10}-za^{-3}-2za^{-5}+2za^{-7}+2za^{-9}-za^{-11}-a^{-2}-a^{-4}-3a^{-6}-2a^{-8}$

Vassiliev invariants

V2 and V3:

(4, 9)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 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 16} 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 128} 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{1160}{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{208}{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 1152} $2288$ 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 384} 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 392} 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{2048}{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 2592} 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{18560}{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{3328}{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{204662}{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{5408}{15}} ${\frac {284288}{45}}$ 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{1450}{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 \frac{13382}{15}}

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 11. 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   1 15               3 1   -2 13             2 1     1 11           3 3       0 9         3 2         1 7       1 3           2 5     2 3             -1 3   1 2               1 1   1                 -1 -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} 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=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}^{2}\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} 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=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}^{2}\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^{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=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, 11]]
Out[2]=	9
In[3]:=	PD[Knot[9, 11]]
Out[3]=	PD[X[1, 4, 2, 5], X[9, 12, 10, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[13, 1, 14, 18], X[5, 15, 6, 14], X[7, 17, 8, 16], X[15, 7, 16, 6], X[17, 9, 18, 8]]
In[4]:=	GaussCode[Knot[9, 11]]
Out[4]=	GaussCode[-1, 4, -3, 1, -6, 8, -7, 9, -2, 3, -4, 2, -5, 6, -8, 7, -9, 5]
In[5]:=	BR[Knot[9, 11]]
Out[5]=	BR[4, {1, 1, 1, 1, -2, 1, 3, -2, 3}]
In[6]:=	alex = Alexander[Knot[9, 11]][t]
Out[6]=	-3 5 7 2 3 7 - t + -- - - - 7 t + 5 t - t 2 t t
In[7]:=	Conway[Knot[9, 11]][z]
Out[7]=	2 4 6 1 + 4 z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 11], Knot[11, NonAlternating, 95]}
In[9]:=	{KnotDet[Knot[9, 11]], KnotSignature[Knot[9, 11]]}
Out[9]=	{33, 4}
In[10]:=	J=Jones[Knot[9, 11]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - 2 q + 3 q - 4 q + 6 q - 5 q + 5 q - 4 q + 2 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 11]}
In[12]:=	A2Invariant[Knot[9, 11]][q]
Out[12]=	8 10 14 16 20 22 26 28 1 - q + 2 q + 2 q + q + q - q - q - q
In[13]:=	Kauffman[Knot[9, 11]][a, z]
Out[13]=	2 2 2 -2 3 -4 -2 z 2 z 2 z 2 z z z 4 z 6 z -- - -- - a - a - --- + --- + --- - --- - -- - --- + ---- + ---- + 8 6 11 9 7 5 3 10 8 6 a a a a a a a a a a 2 2 3 3 3 3 3 4 4 4 5 z 4 z z 3 z 3 z 9 z 8 z 2 z 4 z 7 z ---- + ---- + --- - ---- - ---- + ---- + ---- + ---- - ---- - ---- - 4 2 11 9 7 5 3 10 8 6 a a a a a a a a a a 4 4 5 5 5 5 6 6 6 6 7 5 z 4 z 3 z z 12 z 8 z 3 z z z z 2 z ---- - ---- + ---- - -- - ----- - ---- + ---- + -- - -- + -- + ---- + 4 2 9 7 5 3 8 6 4 2 7 a a a a a a a a a a a 7 7 8 8 4 z 2 z z z ---- + ---- + -- + -- 5 3 6 4 a a a a
In[14]:=	{Vassiliev[2][Knot[9, 11]], Vassiliev[3][Knot[9, 11]]}
Out[14]=	{0, 9}
In[15]:=	Kh[Knot[9, 11]][q, t]
Out[15]=	3 3 5 1 q q 5 7 7 2 9 2 2 q + 2 q + ---- + - + -- + 3 q t + 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 2 q t + 3 q t + 3 q t + 2 q t + q t + 3 q t + 15 6 17 6 19 7 q t + q t + q t