9 39

9_38

9_40

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 39's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 39 at Knotilus!

Knot presentations

Planar diagram presentation	X1627 X3,11,4,10 X7,18,8,1 X17,13,18,12 X9,17,10,16 X5,15,6,14 X15,5,16,4 X11,3,12,2 X13,9,14,8
Gauss code	-1, 8, -2, 7, -6, 1, -3, 9, -5, 2, -8, 4, -9, 6, -7, 5, -4, 3
Dowker-Thistlethwaite code	6 10 14 18 16 2 8 4 12
Conway Notation	[2:2:20]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 12, width is 5, Braid index is 5		[{11, 6}, {2, 7}, {6, 1}, {8, 3}, {5, 2}, {7, 9}, {4, 8}, {10, 5}, {9, 11}, {3, 10}, {1, 4}]

[edit Notes on presentations of 9 39]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 39"];

In[4]:=

PD[K]

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

Out[4]=

X1627 X3,11,4,10 X7,18,8,1 X17,13,18,12 X9,17,10,16 X5,15,6,14 X15,5,16,4 X11,3,12,2 X13,9,14,8

In[5]:=

GaussCode[K]

Out[5]=

-1, 8, -2, 7, -6, 1, -3, 9, -5, 2, -8, 4, -9, 6, -7, 5, -4, 3

In[6]:=

DTCode[K]

Out[6]=

6 10 14 18 16 2 8 4 12

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[2:2:20]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

${\textrm {BR}}(5,\{1,1,2,-1,-3,-2,1,4,3,-2,3,4\})$

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

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

Out[10]=

{ 5, 12, 5 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{11, 6}, {2, 7}, {6, 1}, {8, 3}, {5, 2}, {7, 9}, {4, 8}, {10, 5}, {9, 11}, {3, 10}, {1, 4}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index $\{4,6\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-10] Hyperbolic Volume 12.8103 A-Polynomial See Data:9 39/A-polynomial

[edit Notes for 9 39's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 2

[edit Notes for 9 39's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+14t-21+14t^{-1}-3t^{-2}$
Conway polynomial	$-3z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 55, 2 }
Jones polynomial	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^8+3 q^7-6 q^6+8 q^5-9 q^4+10 q^3-8 q^2+6 q-3+ q^{-1} }
HOMFLY-PT polynomial (db, data sources)	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 -z^4 a^{-2} -2 z^4 a^{-4} +z^2 a^{-2} -3 z^2 a^{-4} +3 z^2 a^{-6} +z^2+2 a^{-2} -2 a^{-4} +2 a^{-6} - a^{-8} }
Kauffman polynomial (db, data sources)	$2z^{8}a^{-4}+2z^{8}a^{-6}+5z^{7}a^{-3}+9z^{7}a^{-5}+4z^{7}a^{-7}+5z^{6}a^{-2}+5z^{6}a^{-4}+3z^{6}a^{-6}+3z^{6}a^{-8}+3z^{5}a^{-1}-7z^{5}a^{-3}-18z^{5}a^{-5}-7z^{5}a^{-7}+z^{5}a^{-9}-7z^{4}a^{-2}-15z^{4}a^{-4}-13z^{4}a^{-6}-6z^{4}a^{-8}+z^{4}-3z^{3}a^{-1}+5z^{3}a^{-3}+12z^{3}a^{-5}+2z^{3}a^{-7}-2z^{3}a^{-9}+5z^{2}a^{-2}+12z^{2}a^{-4}+9z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}-za^{-3}-3za^{-5}-za^{-7}+za^{-9}-2a^{-2}-2a^{-4}-2a^{-6}-a^{-8}$
The A2 invariant	$q^{4}-q^{2}-1+3q^{-2}-q^{-4}+2q^{-6}+q^{-8}-q^{-10}+q^{-12}-2q^{-14}+2q^{-16}-q^{-20}+2q^{-22}-q^{-24}-q^{-26}$
The G2 invariant	$q^{18}-2q^{16}+4q^{14}-6q^{12}+5q^{10}-3q^{8}-2q^{6}+12q^{4}-19q^{2}+28-30q^{-2}+21q^{-4}-3q^{-6}-27q^{-8}+58q^{-10}-76q^{-12}+73q^{-14}-45q^{-16}-6q^{-18}+63q^{-20}-97q^{-22}+101q^{-24}-61q^{-26}+2q^{-28}+53q^{-30}-80q^{-32}+65q^{-34}-12q^{-36}-45q^{-38}+87q^{-40}-83q^{-42}+36q^{-44}+37q^{-46}-103q^{-48}+134q^{-50}-123q^{-52}+66q^{-54}+10q^{-56}-84q^{-58}+131q^{-60}-134q^{-62}+95q^{-64}-29q^{-66}-43q^{-68}+87q^{-70}-93q^{-72}+59q^{-74}-52q^{-78}+80q^{-80}-61q^{-82}+8q^{-84}+57q^{-86}-100q^{-88}+103q^{-90}-65q^{-92}-q^{-94}+60q^{-96}-93q^{-98}+95q^{-100}-63q^{-102}+19q^{-104}+19q^{-106}-45q^{-108}+45q^{-110}-33q^{-112}+17q^{-114}-3q^{-116}-6q^{-118}+8q^{-120}-8q^{-122}+5q^{-124}-2q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 9_39.

A1 Invariants.

Weight	Invariant
1	$q^{3}-2q+3q^{-1}-2q^{-3}+2q^{-5}+q^{-7}-q^{-9}+2q^{-11}-3q^{-13}+2q^{-15}-q^{-17}$
2	$q^{10}-2q^{8}+6q^{4}-8q^{2}-3+18q^{-2}-10q^{-4}-13q^{-6}+21q^{-8}-2q^{-10}-15q^{-12}+11q^{-14}+7q^{-16}-7q^{-18}-6q^{-20}+11q^{-22}+2q^{-24}-18q^{-26}+10q^{-28}+12q^{-30}-20q^{-32}+2q^{-34}+16q^{-36}-11q^{-38}-5q^{-40}+8q^{-42}-q^{-44}-2q^{-46}+q^{-48}$
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 q^{21}-2 q^{19}+3 q^{15}-7 q^{11}-q^9+19 q^7+4 q^5-35 q^3-18 q+50 q^{-1} +49 q^{-3} -58 q^{-5} -81 q^{-7} +48 q^{-9} +110 q^{-11} -22 q^{-13} -124 q^{-15} -8 q^{-17} +120 q^{-19} +35 q^{-21} -92 q^{-23} -56 q^{-25} +63 q^{-27} +66 q^{-29} -28 q^{-31} -69 q^{-33} -7 q^{-35} +70 q^{-37} +34 q^{-39} -68 q^{-41} -65 q^{-43} +66 q^{-45} +87 q^{-47} -51 q^{-49} -109 q^{-51} +28 q^{-53} +119 q^{-55} +3 q^{-57} -116 q^{-59} -33 q^{-61} +92 q^{-63} +58 q^{-65} -59 q^{-67} -66 q^{-69} +28 q^{-71} +54 q^{-73} -35 q^{-77} -11 q^{-79} +17 q^{-81} +8 q^{-83} -5 q^{-85} -4 q^{-87} + q^{-89} +2 q^{-91} - q^{-93} }
4	$q^{36}-2q^{34}+3q^{30}-3q^{28}+q^{26}-5q^{24}+7q^{22}+16q^{20}-16q^{18}-18q^{16}-29q^{14}+34q^{12}+94q^{10}+4q^{8}-88q^{6}-177q^{4}-4q^{2}+269+223q^{-2}-46q^{-4}-458q^{-6}-324q^{-8}+281q^{-10}+587q^{-12}+333q^{-14}-510q^{-16}-757q^{-18}-93q^{-20}+656q^{-22}+779q^{-24}-153q^{-26}-823q^{-28}-512q^{-30}+308q^{-32}+823q^{-34}+256q^{-36}-471q^{-38}-598q^{-40}-94q^{-42}+514q^{-44}+420q^{-46}-67q^{-48}-446q^{-50}-319q^{-52}+170q^{-54}+444q^{-56}+222q^{-58}-308q^{-60}-471q^{-62}-86q^{-64}+492q^{-66}+480q^{-68}-189q^{-70}-631q^{-72}-374q^{-74}+475q^{-76}+743q^{-78}+71q^{-80}-654q^{-82}-709q^{-84}+199q^{-86}+807q^{-88}+460q^{-90}-333q^{-92}-826q^{-94}-250q^{-96}+475q^{-98}+623q^{-100}+160q^{-102}-511q^{-104}-453q^{-106}-18q^{-108}+365q^{-110}+359q^{-112}-71q^{-114}-252q^{-116}-199q^{-118}+31q^{-120}+191q^{-122}+81q^{-124}-19q^{-126}-92q^{-128}-50q^{-130}+31q^{-132}+27q^{-134}+19q^{-136}-11q^{-138}-14q^{-140}+2q^{-142}+q^{-144}+4q^{-146}-q^{-148}-2q^{-150}+q^{-152}$
5	$q^{55}-2q^{53}+3q^{49}-3q^{47}-2q^{45}+3q^{43}+3q^{41}+4q^{39}+3q^{37}-16q^{35}-28q^{33}+q^{31}+45q^{29}+67q^{27}+26q^{25}-78q^{23}-176q^{21}-133q^{19}+107q^{17}+362q^{15}+373q^{13}-4q^{11}-574q^{9}-844q^{7}-376q^{5}+690q^{3}+1486q+1143q^{-1}-420q^{-3}-2090q^{-5}-2322q^{-7}-436q^{-9}+2359q^{-11}+3622q^{-13}+1876q^{-15}-1907q^{-17}-4647q^{-19}-3684q^{-21}+702q^{-23}+4997q^{-25}+5332q^{-27}+1068q^{-29}-4429q^{-31}-6381q^{-33}-2940q^{-35}+3094q^{-37}+6539q^{-39}+4408q^{-41}-1367q^{-43}-5793q^{-45}-5162q^{-47}-316q^{-49}+4443q^{-51}+5172q^{-53}+1581q^{-55}-2899q^{-57}-4539q^{-59}-2348q^{-61}+1461q^{-63}+3655q^{-65}+2656q^{-67}-365q^{-69}-2756q^{-71}-2693q^{-73}-426q^{-75}+2053q^{-77}+2705q^{-79}+986q^{-81}-1633q^{-83}-2831q^{-85}-1472q^{-87}+1364q^{-89}+3173q^{-91}+2078q^{-93}-1183q^{-95}-3652q^{-97}-2847q^{-99}+821q^{-101}+4136q^{-103}+3841q^{-105}-172q^{-107}-4404q^{-109}-4877q^{-111}-889q^{-113}+4209q^{-115}+5774q^{-117}+2249q^{-119}-3374q^{-121}-6210q^{-123}-3713q^{-125}+1949q^{-127}+5923q^{-129}+4882q^{-131}-119q^{-133}-4818q^{-135}-5431q^{-137}-1681q^{-139}+3105q^{-141}+5089q^{-143}+2990q^{-145}-1136q^{-147}-3955q^{-149}-3519q^{-151}-540q^{-153}+2391q^{-155}+3163q^{-157}+1561q^{-159}-845q^{-161}-2237q^{-163}-1830q^{-165}-226q^{-167}+1171q^{-169}+1467q^{-171}+718q^{-173}-315q^{-175}-884q^{-177}-723q^{-179}-124q^{-181}+366q^{-183}+470q^{-185}+238q^{-187}-59q^{-189}-223q^{-191}-183q^{-193}-32q^{-195}+70q^{-197}+84q^{-199}+39q^{-201}-7q^{-203}-33q^{-205}-22q^{-207}+3q^{-209}+8q^{-211}+4q^{-213}+2q^{-215}-q^{-217}-4q^{-219}+q^{-221}+2q^{-223}-q^{-225}$

A2 Invariants.

Weight	Invariant
1,0	$q^{4}-q^{2}-1+3q^{-2}-q^{-4}+2q^{-6}+q^{-8}-q^{-10}+q^{-12}-2q^{-14}+2q^{-16}-q^{-20}+2q^{-22}-q^{-24}-q^{-26}$
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^{12}-4 q^{10}+10 q^8-20 q^6+38 q^4-62 q^2+98-150 q^{-2} +211 q^{-4} -270 q^{-6} +334 q^{-8} -374 q^{-10} +372 q^{-12} -316 q^{-14} +212 q^{-16} -54 q^{-18} -136 q^{-20} +344 q^{-22} -522 q^{-24} +662 q^{-26} -745 q^{-28} +754 q^{-30} -702 q^{-32} +582 q^{-34} -415 q^{-36} +218 q^{-38} -18 q^{-40} -158 q^{-42} +298 q^{-44} -386 q^{-46} +410 q^{-48} -380 q^{-50} +319 q^{-52} -248 q^{-54} +168 q^{-56} -102 q^{-58} +58 q^{-60} -28 q^{-62} +12 q^{-64} -4 q^{-66} + q^{-68} }
2,0	$q^{12}-q^{10}-2q^{8}+2q^{6}+5q^{4}-q^{2}-10+14q^{-4}-11q^{-8}+3q^{-10}+11q^{-12}-q^{-14}-10q^{-16}+3q^{-18}+6q^{-20}-3q^{-22}+2q^{-24}+3q^{-26}-2q^{-28}+8q^{-32}-5q^{-34}-10q^{-36}+2q^{-38}+9q^{-40}-4q^{-42}-12q^{-44}+6q^{-46}+9q^{-48}-2q^{-50}-8q^{-52}+6q^{-56}-3q^{-60}-q^{-62}+q^{-64}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{8}-2q^{6}+5q^{2}-6-q^{-2}+13q^{-4}-10q^{-6}-3q^{-8}+17q^{-10}-10q^{-12}-4q^{-14}+12q^{-16}-4q^{-18}-3q^{-20}+2q^{-22}+5q^{-24}-2q^{-26}-8q^{-28}+8q^{-30}+4q^{-32}-15q^{-34}+9q^{-36}+7q^{-38}-15q^{-40}+7q^{-42}+3q^{-44}-8q^{-46}+4q^{-48}+q^{-50}-2q^{-52}+q^{-54}$
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^5-q^3- q^{-1} +3 q^{-3} - q^{-5} +3 q^{-7} + q^{-9} + q^{-11} - q^{-13} - q^{-15} -2 q^{-19} +2 q^{-21} +2 q^{-25} - q^{-27} +2 q^{-29} - q^{-31} - q^{-33} - q^{-35} }

B2 Invariants.

Weight	Invariant
0,1	$q^{8}-2q^{6}+4q^{4}-7q^{2}+10-13q^{-2}+17q^{-4}-16q^{-6}+15q^{-8}-9q^{-10}+4q^{-12}+4q^{-14}-12q^{-16}+20q^{-18}-27q^{-20}+30q^{-22}-31q^{-24}+28q^{-26}-22q^{-28}+16q^{-30}-6q^{-32}-q^{-34}+9q^{-36}-13q^{-38}+15q^{-40}-17q^{-42}+15q^{-44}-12q^{-46}+8q^{-48}-5q^{-50}+2q^{-52}-q^{-54}$
1,0	$q^{14}-2q^{10}-2q^{8}+2q^{6}+6q^{4}+q^{2}-8-7q^{-2}+6q^{-4}+15q^{-6}+q^{-8}-16q^{-10}-9q^{-12}+13q^{-14}+15q^{-16}-5q^{-18}-16q^{-20}+15q^{-24}+5q^{-26}-12q^{-28}-6q^{-30}+10q^{-32}+9q^{-34}-6q^{-36}-10q^{-38}+4q^{-40}+11q^{-42}-2q^{-44}-12q^{-46}-2q^{-48}+13q^{-50}+6q^{-52}-13q^{-54}-14q^{-56}+9q^{-58}+18q^{-60}-q^{-62}-17q^{-64}-8q^{-66}+12q^{-68}+11q^{-70}-5q^{-72}-10q^{-74}-q^{-76}+6q^{-78}+3q^{-80}-2q^{-82}-2q^{-84}+q^{-88}$

G2 Invariants.

Weight

Invariant

1,0

$q^{18}-2q^{16}+4q^{14}-6q^{12}+5q^{10}-3q^{8}-2q^{6}+12q^{4}-19q^{2}+28-30q^{-2}+21q^{-4}-3q^{-6}-27q^{-8}+58q^{-10}-76q^{-12}+73q^{-14}-45q^{-16}-6q^{-18}+63q^{-20}-97q^{-22}+101q^{-24}-61q^{-26}+2q^{-28}+53q^{-30}-80q^{-32}+65q^{-34}-12q^{-36}-45q^{-38}+87q^{-40}-83q^{-42}+36q^{-44}+37q^{-46}-103q^{-48}+134q^{-50}-123q^{-52}+66q^{-54}+10q^{-56}-84q^{-58}+131q^{-60}-134q^{-62}+95q^{-64}-29q^{-66}-43q^{-68}+87q^{-70}-93q^{-72}+59q^{-74}-52q^{-78}+80q^{-80}-61q^{-82}+8q^{-84}+57q^{-86}-100q^{-88}+103q^{-90}-65q^{-92}-q^{-94}+60q^{-96}-93q^{-98}+95q^{-100}-63q^{-102}+19q^{-104}+19q^{-106}-45q^{-108}+45q^{-110}-33q^{-112}+17q^{-114}-3q^{-116}-6q^{-118}+8q^{-120}-8q^{-122}+5q^{-124}-2q^{-126}+q^{-128}$

.

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 39"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{2}+14t-21+14t^{-1}-3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-3z^{4}+2z^{2}+1$

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]=

{ 55, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

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^8+3 q^7-6 q^6+8 q^5-9 q^4+10 q^3-8 q^2+6 q-3+ q^{-1} }

In[9]:=

HOMFLYPT[K][a, z]

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

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{8}a^{-4}+2z^{8}a^{-6}+5z^{7}a^{-3}+9z^{7}a^{-5}+4z^{7}a^{-7}+5z^{6}a^{-2}+5z^{6}a^{-4}+3z^{6}a^{-6}+3z^{6}a^{-8}+3z^{5}a^{-1}-7z^{5}a^{-3}-18z^{5}a^{-5}-7z^{5}a^{-7}+z^{5}a^{-9}-7z^{4}a^{-2}-15z^{4}a^{-4}-13z^{4}a^{-6}-6z^{4}a^{-8}+z^{4}-3z^{3}a^{-1}+5z^{3}a^{-3}+12z^{3}a^{-5}+2z^{3}a^{-7}-2z^{3}a^{-9}+5z^{2}a^{-2}+12z^{2}a^{-4}+9z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}-za^{-3}-3za^{-5}-za^{-7}+za^{-9}-2a^{-2}-2a^{-4}-2a^{-6}-a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n162,}

Same Jones Polynomial (up to mirroring, 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\leftrightarrow q^{-1}} ): {K11n11, K11n112,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(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 May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 39"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

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

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

Out[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 -3 t^2+14 t-21+14 t^{-1} -3 t^{-2} } , $-q^{8}+3q^{7}-6q^{6}+8q^{5}-9q^{4}+10q^{3}-8q^{2}+6q-3+q^{-1}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{K11n162,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n11, K11n112,}

Vassiliev invariants

V2 and V3:

(2, 4)

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 $8$ $32$ 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 32} 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{460}{3}} ${\frac {116}{3}}$ $256$ ${\frac {2144}{3}}$ ${\frac {320}{3}}$ $192$ 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{256}{3}} $512$ ${\frac {3680}{3}}$ 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{928}{3}} ${\frac {49111}{15}}$ $-{\frac {8524}{15}}$ ${\frac {97084}{45}}$ ${\frac {857}{9}}$ 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{5191}{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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$2 is the signature of 9 39. 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 17                   1 -1 15                 2   2 13               4 1   -3 11             4 2     2 9           5 4       -1 7         5 4         1 5       3 5           2 3     3 5             -2 1   1 4               3 -1   2                 -2 -3 1                   1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{23}-3q^{22}+q^{21}+10q^{20}-16q^{19}-5q^{18}+37q^{17}-30q^{16}-27q^{15}+69q^{14}-32q^{13}-55q^{12}+89q^{11}-23q^{10}-72q^{9}+88q^{8}-9q^{7}-68q^{6}+62q^{5}+4q^{4}-45q^{3}+28q^{2}+7q-17+7q^{-1}+2q^{-2}-3q^{-3}+q^{-4}$
3	$-q^{45}+3q^{44}-q^{43}-5q^{42}-2q^{41}+16q^{40}+8q^{39}-33q^{38}-26q^{37}+51q^{36}+62q^{35}-59q^{34}-120q^{33}+58q^{32}+179q^{31}-25q^{30}-245q^{29}-25q^{28}+298q^{27}+91q^{26}-336q^{25}-162q^{24}+356q^{23}+229q^{22}-357q^{21}-293q^{20}+353q^{19}+331q^{18}-321q^{17}-370q^{16}+291q^{15}+372q^{14}-227q^{13}-373q^{12}+172q^{11}+336q^{10}-100q^{9}-288q^{8}+44q^{7}+220q^{6}+2q^{5}-156q^{4}-18q^{3}+91q^{2}+25q-49-17q^{-1}+23q^{-2}+8q^{-3}-10q^{-4}-2q^{-5}+3q^{-6}+2q^{-7}-3q^{-8}+q^{-9}$
4	$q^{74}-3q^{73}+q^{72}+5q^{71}-3q^{70}+2q^{69}-19q^{68}+4q^{67}+35q^{66}+5q^{65}+6q^{64}-100q^{63}-38q^{62}+108q^{61}+105q^{60}+116q^{59}-260q^{58}-268q^{57}+55q^{56}+286q^{55}+546q^{54}-254q^{53}-651q^{52}-380q^{51}+228q^{50}+1217q^{49}+209q^{48}-799q^{47}-1105q^{46}-348q^{45}+1710q^{44}+1002q^{43}-452q^{42}-1713q^{41}-1256q^{40}+1765q^{39}+1727q^{38}+220q^{37}-1981q^{36}-2105q^{35}+1508q^{34}+2169q^{33}+889q^{32}-1969q^{31}-2683q^{30}+1123q^{29}+2332q^{28}+1419q^{27}-1747q^{26}-2957q^{25}+634q^{24}+2205q^{23}+1798q^{22}-1260q^{21}-2863q^{20}+26q^{19}+1701q^{18}+1925q^{17}-533q^{16}-2296q^{15}-489q^{14}+881q^{13}+1614q^{12}+137q^{11}-1364q^{10}-612q^{9}+132q^{8}+950q^{7}+384q^{6}-521q^{5}-358q^{4}-174q^{3}+345q^{2}+250q-109-89q^{-1}-128q^{-2}+72q^{-3}+77q^{-4}-20q^{-5}+3q^{-6}-38q^{-7}+12q^{-8}+14q^{-9}-9q^{-10}+5q^{-11}-6q^{-12}+3q^{-13}+2q^{-14}-3q^{-15}+q^{-16}$
5	$-q^{110}+3q^{109}-q^{108}-5q^{107}+3q^{106}+3q^{105}+q^{104}+7q^{103}-6q^{102}-30q^{101}-8q^{100}+29q^{99}+47q^{98}+52q^{97}-20q^{96}-132q^{95}-159q^{94}-11q^{93}+211q^{92}+349q^{91}+212q^{90}-236q^{89}-649q^{88}-610q^{87}+50q^{86}+918q^{85}+1245q^{84}+513q^{83}-945q^{82}-2007q^{81}-1554q^{80}+511q^{79}+2637q^{78}+2919q^{77}+657q^{76}-2779q^{75}-4485q^{74}-2468q^{73}+2201q^{72}+5738q^{71}+4783q^{70}-680q^{69}-6469q^{68}-7254q^{67}-1549q^{66}+6351q^{65}+9482q^{64}+4321q^{63}-5428q^{62}-11228q^{61}-7211q^{60}+3854q^{59}+12318q^{58}+9944q^{57}-1903q^{56}-12793q^{55}-12309q^{54}-134q^{53}+12791q^{52}+14176q^{51}+2110q^{50}-12498q^{49}-15624q^{48}-3802q^{47}+11986q^{46}+16645q^{45}+5371q^{44}-11403q^{43}-17433q^{42}-6638q^{41}+10627q^{40}+17843q^{39}+7990q^{38}-9684q^{37}-18085q^{36}-9117q^{35}+8360q^{34}+17780q^{33}+10381q^{32}-6663q^{31}-17086q^{30}-11311q^{29}+4551q^{28}+15589q^{27}+12021q^{26}-2183q^{25}-13495q^{24}-12040q^{23}-208q^{22}+10743q^{21}+11390q^{20}+2243q^{19}-7720q^{18}-9909q^{17}-3653q^{16}+4709q^{15}+7949q^{14}+4195q^{13}-2223q^{12}-5645q^{11}-3988q^{10}+414q^{9}+3563q^{8}+3232q^{7}+517q^{6}-1862q^{5}-2242q^{4}-849q^{3}+768q^{2}+1346q+749-192q^{-1}-679q^{-2}-506q^{-3}-28q^{-4}+280q^{-5}+281q^{-6}+78q^{-7}-109q^{-8}-129q^{-9}-39q^{-10}+25q^{-11}+41q^{-12}+35q^{-13}-11q^{-14}-25q^{-15}+2q^{-16}+3q^{-17}-3q^{-18}+6q^{-19}+q^{-20}-6q^{-21}+3q^{-22}+2q^{-23}-3q^{-24}+q^{-25}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

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