10 65

10_64

10_66

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10 65 Quick Notes

10 65 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X3,10,4,11 X11,19,12,18 X5,15,6,14 X17,7,18,6 X7,17,8,16 X15,9,16,8 X13,1,14,20 X19,13,20,12 X9,2,10,3
Gauss code	-1, 10, -2, 1, -4, 5, -6, 7, -10, 2, -3, 9, -8, 4, -7, 6, -5, 3, -9, 8
Dowker-Thistlethwaite code	4 10 14 16 2 18 20 8 6 12
Conway Notation	[31,3,21]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-10] Hyperbolic Volume 12.0765 A-Polynomial See Data:10 65/A-polynomial

[edit Notes for 10 65's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for 10 65's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-7t^{2}+14t-17+14t^{-1}-7t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+5z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 63, 2 }
Jones polynomial	$-q^{8}+2q^{7}-5q^{6}+8q^{5}-9q^{4}+11q^{3}-10q^{2}+8q-5+3q^{-1}-q^{-2}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}a^{-4}+3z^{4}a^{-2}+4z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+2z^{2}a^{-2}+7z^{2}a^{-4}-3z^{2}a^{-6}-2z^{2}-a^{-2}+5a^{-4}-3a^{-6}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-3}+z^{9}a^{-5}+3z^{8}a^{-2}+6z^{8}a^{-4}+3z^{8}a^{-6}+4z^{7}a^{-1}+5z^{7}a^{-3}+4z^{7}a^{-5}+3z^{7}a^{-7}-4z^{6}a^{-2}-16z^{6}a^{-4}-7z^{6}a^{-6}+2z^{6}a^{-8}+3z^{6}+az^{5}-9z^{5}a^{-1}-17z^{5}a^{-3}-14z^{5}a^{-5}-6z^{5}a^{-7}+z^{5}a^{-9}+z^{4}a^{-2}+24z^{4}a^{-4}+12z^{4}a^{-6}-4z^{4}a^{-8}-7z^{4}-2az^{3}+6z^{3}a^{-1}+20z^{3}a^{-3}+19z^{3}a^{-5}+4z^{3}a^{-7}-3z^{3}a^{-9}-z^{2}a^{-2}-17z^{2}a^{-4}-12z^{2}a^{-6}+z^{2}a^{-8}+3z^{2}-2za^{-1}-6za^{-3}-8za^{-5}-2za^{-7}+2za^{-9}+a^{-2}+5a^{-4}+3a^{-6}$
The A2 invariant	$-q^{6}+q^{4}+2q^{-2}-3q^{-4}+q^{-6}+2q^{-10}+4q^{-12}+2q^{-16}-2q^{-18}-2q^{-20}-q^{-24}$
The G2 invariant	$q^{32}-2q^{30}+4q^{28}-7q^{26}+6q^{24}-5q^{22}-2q^{20}+14q^{18}-23q^{16}+32q^{14}-32q^{12}+19q^{10}+2q^{8}-34q^{6}+62q^{4}-74q^{2}+66-35q^{-2}-10q^{-4}+61q^{-6}-92q^{-8}+96q^{-10}-66q^{-12}+10q^{-14}+41q^{-16}-75q^{-18}+71q^{-20}-34q^{-22}-13q^{-24}+60q^{-26}-74q^{-28}+46q^{-30}+8q^{-32}-78q^{-34}+118q^{-36}-116q^{-38}+74q^{-40}-75q^{-44}+136q^{-46}-141q^{-48}+113q^{-50}-48q^{-52}-31q^{-54}+93q^{-56}-103q^{-58}+85q^{-60}-32q^{-62}-18q^{-64}+62q^{-66}-63q^{-68}+30q^{-70}+15q^{-72}-66q^{-74}+88q^{-76}-72q^{-78}+17q^{-80}+38q^{-82}-82q^{-84}+99q^{-86}-84q^{-88}+41q^{-90}+2q^{-92}-47q^{-94}+64q^{-96}-62q^{-98}+43q^{-100}-17q^{-102}-3q^{-104}+16q^{-106}-23q^{-108}+21q^{-110}-15q^{-112}+8q^{-114}-q^{-116}-3q^{-118}+4q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 10_65.

A1 Invariants.

Weight	Invariant
1	$-q^{5}+2q^{3}-2q+3q^{-1}-2q^{-3}+q^{-5}+2q^{-7}-q^{-9}+3q^{-11}-3q^{-13}+q^{-15}-q^{-17}$
2	$q^{16}-2q^{14}-q^{12}+6q^{10}-5q^{8}-6q^{6}+13q^{4}-4q^{2}-14+18q^{-2}+3q^{-4}-18q^{-6}+13q^{-8}+9q^{-10}-12q^{-12}-2q^{-14}+8q^{-16}+2q^{-18}-14q^{-20}+6q^{-22}+15q^{-24}-17q^{-26}-q^{-28}+18q^{-30}-13q^{-32}-7q^{-34}+12q^{-36}-4q^{-38}-5q^{-40}+4q^{-42}-q^{-46}+q^{-48}$
3	$-q^{33}+2q^{31}+q^{29}-3q^{27}-3q^{25}+5q^{23}+8q^{21}-9q^{19}-14q^{17}+10q^{15}+24q^{13}-10q^{11}-38q^{9}+2q^{7}+55q^{5}+12q^{3}-66q-36q^{-1}+73q^{-3}+61q^{-5}-64q^{-7}-81q^{-9}+49q^{-11}+95q^{-13}-28q^{-15}-92q^{-17}+5q^{-19}+80q^{-21}+22q^{-23}-61q^{-25}-37q^{-27}+36q^{-29}+55q^{-31}-13q^{-33}-71q^{-35}-15q^{-37}+77q^{-39}+38q^{-41}-81q^{-43}-57q^{-45}+71q^{-47}+78q^{-49}-55q^{-51}-83q^{-53}+30q^{-55}+84q^{-57}-10q^{-59}-68q^{-61}-13q^{-63}+50q^{-65}+20q^{-67}-31q^{-69}-19q^{-71}+13q^{-73}+15q^{-75}-5q^{-77}-7q^{-79}+2q^{-81}+4q^{-83}-q^{-85}-q^{-87}+q^{-91}-q^{-93}$
4	$q^{56}-2q^{54}-q^{52}+3q^{50}+3q^{46}-8q^{44}-3q^{42}+11q^{40}+q^{38}+9q^{36}-24q^{34}-16q^{32}+28q^{30}+19q^{28}+27q^{26}-60q^{24}-64q^{22}+32q^{20}+71q^{18}+112q^{16}-73q^{14}-178q^{12}-72q^{10}+95q^{8}+300q^{6}+60q^{4}-255q^{2}-315-64q^{-2}+446q^{-4}+356q^{-6}-115q^{-8}-502q^{-10}-383q^{-12}+345q^{-14}+570q^{-16}+186q^{-18}-433q^{-20}-589q^{-22}+71q^{-24}+510q^{-26}+385q^{-28}-185q^{-30}-526q^{-32}-165q^{-34}+274q^{-36}+395q^{-38}+49q^{-40}-317q^{-42}-285q^{-44}+40q^{-46}+322q^{-48}+228q^{-50}-102q^{-52}-382q^{-54}-182q^{-56}+250q^{-58}+402q^{-60}+112q^{-62}-449q^{-64}-408q^{-66}+113q^{-68}+522q^{-70}+369q^{-72}-366q^{-74}-557q^{-76}-139q^{-78}+439q^{-80}+558q^{-82}-95q^{-84}-478q^{-86}-363q^{-88}+146q^{-90}+501q^{-92}+162q^{-94}-190q^{-96}-349q^{-98}-119q^{-100}+238q^{-102}+198q^{-104}+48q^{-106}-159q^{-108}-154q^{-110}+31q^{-112}+78q^{-114}+85q^{-116}-16q^{-118}-63q^{-120}-15q^{-122}+33q^{-126}+7q^{-128}-12q^{-130}-q^{-132}-7q^{-134}+6q^{-136}+q^{-138}-3q^{-140}+2q^{-142}-2q^{-144}+q^{-146}-q^{-150}+q^{-152}$
5	$-q^{85}+2q^{83}+q^{81}-3q^{79}+3q^{71}+2q^{69}-7q^{67}-5q^{65}+7q^{63}+9q^{61}+5q^{59}-6q^{57}-22q^{55}-23q^{53}+14q^{51}+56q^{49}+46q^{47}-11q^{45}-88q^{43}-116q^{41}-32q^{39}+133q^{37}+227q^{35}+127q^{33}-122q^{31}-353q^{29}-341q^{27}+7q^{25}+464q^{23}+634q^{21}+284q^{19}-435q^{17}-972q^{15}-779q^{13}+162q^{11}+1207q^{9}+1436q^{7}+421q^{5}-1191q^{3}-2079q-1306q^{-1}+775q^{-3}+2547q^{-5}+2326q^{-7}+22q^{-9}-2599q^{-11}-3241q^{-13}-1110q^{-15}+2199q^{-17}+3837q^{-19}+2214q^{-21}-1405q^{-23}-3926q^{-25}-3100q^{-27}+405q^{-29}+3543q^{-31}+3593q^{-33}+548q^{-35}-2816q^{-37}-3597q^{-39}-1292q^{-41}+1927q^{-43}+3245q^{-45}+1733q^{-47}-1090q^{-49}-2682q^{-51}-1863q^{-53}+385q^{-55}+2048q^{-57}+1855q^{-59}+141q^{-61}-1532q^{-63}-1778q^{-65}-542q^{-67}+1093q^{-69}+1789q^{-71}+931q^{-73}-783q^{-75}-1903q^{-77}-1373q^{-79}+485q^{-81}+2095q^{-83}+1935q^{-85}-93q^{-87}-2293q^{-89}-2577q^{-91}-448q^{-93}+2332q^{-95}+3215q^{-97}+1187q^{-99}-2117q^{-101}-3711q^{-103}-2022q^{-105}+1545q^{-107}+3878q^{-109}+2850q^{-111}-692q^{-113}-3607q^{-115}-3422q^{-117}-365q^{-119}+2908q^{-121}+3623q^{-123}+1317q^{-125}-1862q^{-127}-3303q^{-129}-2033q^{-131}+724q^{-133}+2618q^{-135}+2259q^{-137}+258q^{-139}-1652q^{-141}-2071q^{-143}-910q^{-145}+739q^{-147}+1557q^{-149}+1111q^{-151}-28q^{-153}-927q^{-155}-994q^{-157}-363q^{-159}+393q^{-161}+693q^{-163}+445q^{-165}-32q^{-167}-367q^{-169}-368q^{-171}-116q^{-173}+140q^{-175}+221q^{-177}+132q^{-179}-16q^{-181}-105q^{-183}-87q^{-185}-25q^{-187}+34q^{-189}+48q^{-191}+19q^{-193}-9q^{-195}-14q^{-197}-12q^{-199}-5q^{-201}+9q^{-203}+5q^{-205}-2q^{-207}+q^{-209}-3q^{-213}+q^{-215}+2q^{-217}-q^{-219}+q^{-223}-q^{-225}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{6}+q^{4}+2q^{-2}-3q^{-4}+q^{-6}+2q^{-10}+4q^{-12}+2q^{-16}-2q^{-18}-2q^{-20}-q^{-24}$
1,1	$q^{20}-4q^{18}+10q^{16}-22q^{14}+44q^{12}-74q^{10}+112q^{8}-166q^{6}+226q^{4}-282q^{2}+328-366q^{-2}+369q^{-4}-324q^{-6}+240q^{-8}-110q^{-10}-46q^{-12}+240q^{-14}-418q^{-16}+580q^{-18}-691q^{-20}+752q^{-22}-746q^{-24}+678q^{-26}-560q^{-28}+392q^{-30}-210q^{-32}+32q^{-34}+125q^{-36}-254q^{-38}+340q^{-40}-370q^{-42}+357q^{-44}-326q^{-46}+272q^{-48}-210q^{-50}+151q^{-52}-106q^{-54}+68q^{-56}-42q^{-58}+25q^{-60}-12q^{-62}+6q^{-64}-2q^{-66}+q^{-68}$
2,0	$q^{18}-q^{16}-2q^{14}+2q^{12}+3q^{10}-3q^{8}-5q^{6}+4q^{4}+5q^{2}-7-6q^{-2}+8q^{-4}+2q^{-6}-6q^{-8}+5q^{-10}+11q^{-12}+2q^{-14}-3q^{-16}+3q^{-18}-2q^{-20}-11q^{-22}+q^{-24}+5q^{-26}-4q^{-28}+q^{-30}+14q^{-32}+6q^{-34}-5q^{-36}-2q^{-38}+3q^{-40}-5q^{-42}-11q^{-44}-2q^{-46}+2q^{-48}-q^{-50}-q^{-52}+q^{-54}+2q^{-56}+q^{-58}+q^{-62}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{14}-2q^{12}+3q^{8}-6q^{6}+3q^{4}+8q^{2}-11+3q^{-2}+11q^{-4}-17q^{-6}-q^{-8}+12q^{-10}-10q^{-12}+11q^{-16}+4q^{-18}+q^{-20}+2q^{-22}+12q^{-24}-3q^{-26}-11q^{-28}+11q^{-30}-3q^{-32}-18q^{-34}+10q^{-36}-12q^{-40}+7q^{-42}+q^{-44}-4q^{-46}+3q^{-48}+q^{-50}-q^{-52}+q^{-54}$
1,0,0	$-q^{7}+q^{5}-q^{3}+2q-q^{-1}+2q^{-3}-3q^{-5}-q^{-9}+q^{-11}+3q^{-13}+3q^{-15}+5q^{-17}+2q^{-21}-3q^{-23}-q^{-25}-3q^{-27}-q^{-31}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{16}-q^{14}-q^{12}+q^{10}-q^{8}-2q^{6}+3q^{4}+4q^{2}-3-2q^{-2}+6q^{-4}+2q^{-6}-13q^{-8}-2q^{-10}+10q^{-12}-6q^{-14}-11q^{-16}+8q^{-18}+7q^{-20}-8q^{-22}+2q^{-24}+16q^{-26}+7q^{-28}+2q^{-30}+19q^{-32}+15q^{-34}-8q^{-36}+5q^{-40}-15q^{-42}-21q^{-44}-3q^{-46}-q^{-48}-11q^{-50}-5q^{-52}+7q^{-54}+4q^{-56}-3q^{-58}+3q^{-60}+4q^{-62}+q^{-68}$
1,0,0,0	$-q^{8}+q^{6}-q^{4}+q^{2}+1-q^{-2}+2q^{-4}-3q^{-6}-2q^{-10}+q^{-14}+3q^{-16}+4q^{-18}+4q^{-20}+5q^{-22}+2q^{-26}-3q^{-28}-2q^{-30}-2q^{-32}-3q^{-34}-q^{-38}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{14}+2q^{12}-4q^{10}+7q^{8}-10q^{6}+13q^{4}-16q^{2}+17-17q^{-2}+15q^{-4}-9q^{-6}+3q^{-8}+6q^{-10}-14q^{-12}+24q^{-14}-29q^{-16}+34q^{-18}-33q^{-20}+32q^{-22}-26q^{-24}+19q^{-26}-9q^{-28}+q^{-30}+7q^{-32}-12q^{-34}+16q^{-36}-18q^{-38}+16q^{-40}-15q^{-42}+11q^{-44}-8q^{-46}+5q^{-48}-3q^{-50}+q^{-52}-q^{-54}$
1,0	$q^{24}-2q^{20}-2q^{18}+2q^{16}+5q^{14}-q^{12}-8q^{10}-4q^{8}+9q^{6}+12q^{4}-4q^{2}-16-5q^{-2}+15q^{-4}+14q^{-6}-10q^{-8}-19q^{-10}-q^{-12}+17q^{-14}+7q^{-16}-12q^{-18}-9q^{-20}+10q^{-22}+12q^{-24}-4q^{-26}-10q^{-28}+5q^{-30}+13q^{-32}+q^{-34}-11q^{-36}-q^{-38}+13q^{-40}+7q^{-42}-12q^{-44}-11q^{-46}+9q^{-48}+16q^{-50}-4q^{-52}-20q^{-54}-9q^{-56}+14q^{-58}+13q^{-60}-8q^{-62}-16q^{-64}-3q^{-66}+11q^{-68}+6q^{-70}-4q^{-72}-6q^{-74}+4q^{-78}+2q^{-80}-q^{-82}-q^{-84}+q^{-88}$

D4 Invariants.

Weight

Invariant

1,0,0,0

$q^{18}-2q^{16}+2q^{14}-4q^{12}+6q^{10}-8q^{8}+9q^{6}-10q^{4}+14q^{2}-13+13q^{-2}-12q^{-4}+12q^{-6}-11q^{-8}+q^{-10}-3q^{-12}-4q^{-14}+8q^{-16}-16q^{-18}+19q^{-20}-17q^{-22}+32q^{-24}-20q^{-26}+31q^{-28}-19q^{-30}+29q^{-32}-15q^{-34}+14q^{-36}-14q^{-38}+2q^{-40}-2q^{-42}-8q^{-44}+2q^{-46}-14q^{-48}+12q^{-50}-14q^{-52}+12q^{-54}-14q^{-56}+12q^{-58}-9q^{-60}+7q^{-62}-6q^{-64}+5q^{-66}-2q^{-68}+2q^{-70}-q^{-72}+q^{-74}$

G2 Invariants.

Weight

Invariant

1,0

$q^{32}-2q^{30}+4q^{28}-7q^{26}+6q^{24}-5q^{22}-2q^{20}+14q^{18}-23q^{16}+32q^{14}-32q^{12}+19q^{10}+2q^{8}-34q^{6}+62q^{4}-74q^{2}+66-35q^{-2}-10q^{-4}+61q^{-6}-92q^{-8}+96q^{-10}-66q^{-12}+10q^{-14}+41q^{-16}-75q^{-18}+71q^{-20}-34q^{-22}-13q^{-24}+60q^{-26}-74q^{-28}+46q^{-30}+8q^{-32}-78q^{-34}+118q^{-36}-116q^{-38}+74q^{-40}-75q^{-44}+136q^{-46}-141q^{-48}+113q^{-50}-48q^{-52}-31q^{-54}+93q^{-56}-103q^{-58}+85q^{-60}-32q^{-62}-18q^{-64}+62q^{-66}-63q^{-68}+30q^{-70}+15q^{-72}-66q^{-74}+88q^{-76}-72q^{-78}+17q^{-80}+38q^{-82}-82q^{-84}+99q^{-86}-84q^{-88}+41q^{-90}+2q^{-92}-47q^{-94}+64q^{-96}-62q^{-98}+43q^{-100}-17q^{-102}-3q^{-104}+16q^{-106}-23q^{-108}+21q^{-110}-15q^{-112}+8q^{-114}-q^{-116}-3q^{-118}+4q^{-120}-4q^{-122}+3q^{-124}-q^{-126}+q^{-128}$

.

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["10 65"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-7t^{2}+14t-17+14t^{-1}-7t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+5z^{4}+4z^{2}+1$

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]=

$\left\{2,t^{2}+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 63, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+2q^{7}-5q^{6}+8q^{5}-9q^{4}+11q^{3}-10q^{2}+8q-5+3q^{-1}-q^{-2}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+z^{6}a^{-4}+3z^{4}a^{-2}+4z^{4}a^{-4}-z^{4}a^{-6}-z^{4}+2z^{2}a^{-2}+7z^{2}a^{-4}-3z^{2}a^{-6}-2z^{2}-a^{-2}+5a^{-4}-3a^{-6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-3}+z^{9}a^{-5}+3z^{8}a^{-2}+6z^{8}a^{-4}+3z^{8}a^{-6}+4z^{7}a^{-1}+5z^{7}a^{-3}+4z^{7}a^{-5}+3z^{7}a^{-7}-4z^{6}a^{-2}-16z^{6}a^{-4}-7z^{6}a^{-6}+2z^{6}a^{-8}+3z^{6}+az^{5}-9z^{5}a^{-1}-17z^{5}a^{-3}-14z^{5}a^{-5}-6z^{5}a^{-7}+z^{5}a^{-9}+z^{4}a^{-2}+24z^{4}a^{-4}+12z^{4}a^{-6}-4z^{4}a^{-8}-7z^{4}-2az^{3}+6z^{3}a^{-1}+20z^{3}a^{-3}+19z^{3}a^{-5}+4z^{3}a^{-7}-3z^{3}a^{-9}-z^{2}a^{-2}-17z^{2}a^{-4}-12z^{2}a^{-6}+z^{2}a^{-8}+3z^{2}-2za^{-1}-6za^{-3}-8za^{-5}-2za^{-7}+2za^{-9}+a^{-2}+5a^{-4}+3a^{-6}$

Vassiliev invariants

V2 and V3:

(4, 7)

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 $16$ $56$ $128$ ${\frac {776}{3}}$ ${\frac {64}{3}}$ $896$ ${\frac {4304}{3}}$ ${\frac {704}{3}}$ $152$ ${\frac {2048}{3}}$ $1568$ ${\frac {12416}{3}}$ ${\frac {1024}{3}}$ ${\frac {118262}{15}}$ ${\frac {2624}{5}}$ ${\frac {107888}{45}}$ ${\frac {1018}{9}}$ ${\frac {3062}{15}}$

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 10 65. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-3	-2	-1	0	1	2	3	4	5	6	7	χ
17											1	-1
15										1		1
13									4	1		-3
11								4	1			3
9							5	4				-1
7						6	4					2
5					4	5						1
3				4	6							-2
1			2	5								3
-1		1	3									-2
-3		2										2
-5	1											-1

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 65]]
Out[2]=	10
In[3]:=	PD[Knot[10, 65]]
Out[3]=	PD[X[1, 4, 2, 5], X[3, 10, 4, 11], X[11, 19, 12, 18], X[5, 15, 6, 14], X[17, 7, 18, 6], X[7, 17, 8, 16], X[15, 9, 16, 8], X[13, 1, 14, 20], X[19, 13, 20, 12], X[9, 2, 10, 3]]
In[4]:=	GaussCode[Knot[10, 65]]
Out[4]=	GaussCode[-1, 10, -2, 1, -4, 5, -6, 7, -10, 2, -3, 9, -8, 4, -7, 6, -5, 3, -9, 8]
In[5]:=	BR[Knot[10, 65]]
Out[5]=	BR[4, {1, 1, 2, -1, 2, -3, 2, 2, 2, -3, -3}]
In[6]:=	alex = Alexander[Knot[10, 65]][t]
Out[6]=	2 7 14 2 3 -17 + -- - -- + -- + 14 t - 7 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[10, 65]][z]
Out[7]=	2 4 6 1 + 4 z + 5 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 65], Knot[10, 77], Knot[11, NonAlternating, 71], Knot[11, NonAlternating, 75]}
In[9]:=	{KnotDet[Knot[10, 65]], KnotSignature[Knot[10, 65]]}
Out[9]=	{63, 2}
In[10]:=	J=Jones[Knot[10, 65]][q]
Out[10]=	-2 3 2 3 4 5 6 7 8 -5 - q + - + 8 q - 10 q + 11 q - 9 q + 8 q - 5 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 65]}
In[12]:=	A2Invariant[Knot[10, 65]][q]
Out[12]=	-6 -4 2 4 6 10 12 16 18 20 -q + q + 2 q - 3 q + q + 2 q + 4 q + 2 q - 2 q - 2 q - 24 q
In[13]:=	Kauffman[Knot[10, 65]][a, z]
Out[13]=	2 2 3 5 -2 2 z 2 z 8 z 6 z 2 z 2 z 12 z -- + -- + a + --- - --- - --- - --- - --- + 3 z + -- - ----- - 6 4 9 7 5 3 a 8 6 a a a a a a a a 2 2 3 3 3 3 3 17 z z 3 z 4 z 19 z 20 z 6 z 3 4 ----- - -- - ---- + ---- + ----- + ----- + ---- - 2 a z - 7 z - 4 2 9 7 5 3 a a a a a a a 4 4 4 4 5 5 5 5 5 4 z 12 z 24 z z z 6 z 14 z 17 z 9 z 5 ---- + ----- + ----- + -- + -- - ---- - ----- - ----- - ---- + a z + 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 7 7 6 2 z 7 z 16 z 4 z 3 z 4 z 5 z 4 z 3 z + ---- - ---- - ----- - ---- + ---- + ---- + ---- + ---- + 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 3 z 6 z 3 z z z ---- + ---- + ---- + -- + -- 6 4 2 5 3 a a a a a
In[14]:=	{Vassiliev[2][Knot[10, 65]], Vassiliev[3][Knot[10, 65]]}
Out[14]=	{0, 7}
In[15]:=	Kh[Knot[10, 65]][q, t]
Out[15]=	3 1 2 1 3 2 q 3 5 5 q + 4 q + ----- + ----- + ---- + --- + --- + 6 q t + 4 q t + 5 3 3 2 2 q t t q t q t q t 5 2 7 2 7 3 9 3 9 4 11 4 11 5 5 q t + 6 q t + 4 q t + 5 q t + 4 q t + 4 q t + q t + 13 5 13 6 15 6 17 7 4 q t + q t + q t + q t