10 33

10_32

10_34

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10 33 Quick Notes

10 33 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X14,6,15,5 X20,15,1,16 X16,7,17,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X2,14,3,13 X12,4,13,3 X4,12,5,11
Gauss code	1, -8, 9, -10, 2, -1, 4, -5, 6, -7, 10, -9, 8, -2, 3, -4, 7, -6, 5, -3
Dowker-Thistlethwaite code	6 12 14 16 18 4 2 20 10 8
Conway Notation	[311113]

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 11.5357 A-Polynomial See Data:10 33/A-polynomial

[edit Notes for 10 33's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

[edit Notes for 10 33's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t^{2}-16t+25-16t^{-1}+4t^{-2}$
Conway polynomial	$4z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 65, 0 }
Jones polynomial	$-q^{5}+3q^{4}-5q^{3}+8q^{2}-10q+11-10q^{-1}+8q^{-2}-5q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{4}+z^{4}a^{2}+2z^{4}+2z^{2}+1+z^{4}a^{-2}-z^{2}a^{-4}$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+3z^{8}a^{-2}+6z^{8}+4a^{3}z^{7}+5az^{7}+5z^{7}a^{-1}+4z^{7}a^{-3}+3a^{4}z^{6}-4a^{2}z^{6}-4z^{6}a^{-2}+3z^{6}a^{-4}-14z^{6}+a^{5}z^{5}-9a^{3}z^{5}-16az^{5}-16z^{5}a^{-1}-9z^{5}a^{-3}+z^{5}a^{-5}-7a^{4}z^{4}+a^{2}z^{4}+z^{4}a^{-2}-7z^{4}a^{-4}+16z^{4}-2a^{5}z^{3}+6a^{3}z^{3}+18az^{3}+18z^{3}a^{-1}+6z^{3}a^{-3}-2z^{3}a^{-5}+3a^{4}z^{2}+3z^{2}a^{-4}-6z^{2}-2a^{3}z-6az-6za^{-1}-2za^{-3}+1$
The A2 invariant	$-q^{16}+q^{14}+q^{12}-2q^{10}+2q^{8}-q^{4}+2q^{2}-1+2q^{-2}-q^{-4}+2q^{-8}-2q^{-10}+q^{-12}+q^{-14}-q^{-16}$
The G2 invariant	$q^{80}-2q^{78}+4q^{76}-7q^{74}+6q^{72}-5q^{70}-2q^{68}+14q^{66}-23q^{64}+32q^{62}-32q^{60}+19q^{58}+3q^{56}-33q^{54}+60q^{52}-73q^{50}+67q^{48}-37q^{46}-9q^{44}+57q^{42}-88q^{40}+96q^{38}-70q^{36}+20q^{34}+31q^{32}-69q^{30}+73q^{28}-46q^{26}+45q^{22}-65q^{20}+47q^{18}-3q^{16}-55q^{14}+102q^{12}-111q^{10}+80q^{8}-14q^{6}-61q^{4}+124q^{2}-145+124q^{-2}-61q^{-4}-14q^{-6}+80q^{-8}-111q^{-10}+102q^{-12}-55q^{-14}-3q^{-16}+47q^{-18}-65q^{-20}+45q^{-22}-46q^{-26}+73q^{-28}-69q^{-30}+31q^{-32}+20q^{-34}-70q^{-36}+96q^{-38}-88q^{-40}+57q^{-42}-9q^{-44}-37q^{-46}+67q^{-48}-73q^{-50}+60q^{-52}-33q^{-54}+3q^{-56}+19q^{-58}-32q^{-60}+32q^{-62}-23q^{-64}+14q^{-66}-2q^{-68}-5q^{-70}+6q^{-72}-7q^{-74}+4q^{-76}-2q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_33.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+2q^{9}-2q^{7}+3q^{5}-2q^{3}+q+q^{-1}-2q^{-3}+3q^{-5}-2q^{-7}+2q^{-9}-q^{-11}$
2	$q^{32}-2q^{30}-q^{28}+6q^{26}-5q^{24}-6q^{22}+13q^{20}-4q^{18}-13q^{16}+18q^{14}+q^{12}-18q^{10}+13q^{8}+6q^{6}-13q^{4}+q^{2}+9+q^{-2}-13q^{-4}+6q^{-6}+13q^{-8}-18q^{-10}+q^{-12}+18q^{-14}-13q^{-16}-4q^{-18}+13q^{-20}-6q^{-22}-5q^{-24}+6q^{-26}-q^{-28}-2q^{-30}+q^{-32}$
3	$-q^{63}+2q^{61}+q^{59}-3q^{57}-3q^{55}+5q^{53}+8q^{51}-9q^{49}-14q^{47}+10q^{45}+23q^{43}-10q^{41}-35q^{39}+4q^{37}+51q^{35}+6q^{33}-61q^{31}-25q^{29}+69q^{27}+45q^{25}-66q^{23}-63q^{21}+56q^{19}+76q^{17}-42q^{15}-78q^{13}+20q^{11}+73q^{9}+q^{7}-61q^{5}-21q^{3}+44q+44q^{-1}-21q^{-3}-61q^{-5}+q^{-7}+73q^{-9}+20q^{-11}-78q^{-13}-42q^{-15}+76q^{-17}+56q^{-19}-63q^{-21}-66q^{-23}+45q^{-25}+69q^{-27}-25q^{-29}-61q^{-31}+6q^{-33}+51q^{-35}+4q^{-37}-35q^{-39}-10q^{-41}+23q^{-43}+10q^{-45}-14q^{-47}-9q^{-49}+8q^{-51}+5q^{-53}-3q^{-55}-3q^{-57}+q^{-59}+2q^{-61}-q^{-63}$
4	$q^{104}-2q^{102}-q^{100}+3q^{98}+3q^{94}-8q^{92}-3q^{90}+11q^{88}+q^{86}+9q^{84}-24q^{82}-15q^{80}+28q^{78}+16q^{76}+24q^{74}-58q^{72}-54q^{70}+38q^{68}+61q^{66}+91q^{64}-82q^{62}-151q^{60}-31q^{58}+98q^{56}+245q^{54}-q^{52}-237q^{50}-218q^{48}+6q^{46}+393q^{44}+216q^{42}-179q^{40}-398q^{38}-220q^{36}+375q^{34}+411q^{32}+21q^{30}-408q^{28}-406q^{26}+200q^{24}+430q^{22}+202q^{20}-256q^{18}-418q^{16}-3q^{14}+297q^{12}+276q^{10}-65q^{8}-310q^{6}-165q^{4}+118q^{2}+283+118q^{-2}-165q^{-4}-310q^{-6}-65q^{-8}+276q^{-10}+297q^{-12}-3q^{-14}-418q^{-16}-256q^{-18}+202q^{-20}+430q^{-22}+200q^{-24}-406q^{-26}-408q^{-28}+21q^{-30}+411q^{-32}+375q^{-34}-220q^{-36}-398q^{-38}-179q^{-40}+216q^{-42}+393q^{-44}+6q^{-46}-218q^{-48}-237q^{-50}-q^{-52}+245q^{-54}+98q^{-56}-31q^{-58}-151q^{-60}-82q^{-62}+91q^{-64}+61q^{-66}+38q^{-68}-54q^{-70}-58q^{-72}+24q^{-74}+16q^{-76}+28q^{-78}-15q^{-80}-24q^{-82}+9q^{-84}+q^{-86}+11q^{-88}-3q^{-90}-8q^{-92}+3q^{-94}+3q^{-98}-q^{-100}-2q^{-102}+q^{-104}$
5	$-q^{155}+2q^{153}+q^{151}-3q^{149}+3q^{141}+2q^{139}-7q^{137}-5q^{135}+7q^{133}+9q^{131}+5q^{129}-7q^{127}-22q^{125}-20q^{123}+17q^{121}+55q^{119}+38q^{117}-21q^{115}-89q^{113}-98q^{111}-4q^{109}+142q^{107}+197q^{105}+68q^{103}-161q^{101}-321q^{99}-234q^{97}+111q^{95}+454q^{93}+475q^{91}+74q^{89}-504q^{87}-780q^{85}-421q^{83}+393q^{81}+1046q^{79}+927q^{77}-58q^{75}-1179q^{73}-1467q^{71}-522q^{69}+1052q^{67}+1955q^{65}+1250q^{63}-658q^{61}-2208q^{59}-1983q^{57}+6q^{55}+2170q^{53}+2578q^{51}+734q^{49}-1836q^{47}-2882q^{45}-1428q^{43}+1275q^{41}+2871q^{39}+1948q^{37}-653q^{35}-2575q^{33}-2178q^{31}+55q^{29}+2088q^{27}+2182q^{25}+404q^{23}-1548q^{21}-1986q^{19}-711q^{17}+1020q^{15}+1708q^{13}+910q^{11}-563q^{9}-1444q^{7}-1048q^{5}+181q^{3}+1215q+1215q^{-1}+181q^{-3}-1048q^{-5}-1444q^{-7}-563q^{-9}+910q^{-11}+1708q^{-13}+1020q^{-15}-711q^{-17}-1986q^{-19}-1548q^{-21}+404q^{-23}+2182q^{-25}+2088q^{-27}+55q^{-29}-2178q^{-31}-2575q^{-33}-653q^{-35}+1948q^{-37}+2871q^{-39}+1275q^{-41}-1428q^{-43}-2882q^{-45}-1836q^{-47}+734q^{-49}+2578q^{-51}+2170q^{-53}+6q^{-55}-1983q^{-57}-2208q^{-59}-658q^{-61}+1250q^{-63}+1955q^{-65}+1052q^{-67}-522q^{-69}-1467q^{-71}-1179q^{-73}-58q^{-75}+927q^{-77}+1046q^{-79}+393q^{-81}-421q^{-83}-780q^{-85}-504q^{-87}+74q^{-89}+475q^{-91}+454q^{-93}+111q^{-95}-234q^{-97}-321q^{-99}-161q^{-101}+68q^{-103}+197q^{-105}+142q^{-107}-4q^{-109}-98q^{-111}-89q^{-113}-21q^{-115}+38q^{-117}+55q^{-119}+17q^{-121}-20q^{-123}-22q^{-125}-7q^{-127}+5q^{-129}+9q^{-131}+7q^{-133}-5q^{-135}-7q^{-137}+2q^{-139}+3q^{-141}-3q^{-149}+q^{-151}+2q^{-153}-q^{-155}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+q^{12}-2q^{10}+2q^{8}-q^{4}+2q^{2}-1+2q^{-2}-q^{-4}+2q^{-8}-2q^{-10}+q^{-12}+q^{-14}-q^{-16}$
1,1	$q^{44}-4q^{42}+10q^{40}-22q^{38}+44q^{36}-74q^{34}+112q^{32}-166q^{30}+222q^{28}-278q^{26}+330q^{24}-362q^{22}+371q^{20}-340q^{18}+270q^{16}-158q^{14}+8q^{12}+164q^{10}-340q^{8}+510q^{6}-645q^{4}+732q^{2}-762+732q^{-2}-645q^{-4}+510q^{-6}-340q^{-8}+164q^{-10}+8q^{-12}-158q^{-14}+270q^{-16}-340q^{-18}+371q^{-20}-362q^{-22}+330q^{-24}-278q^{-26}+222q^{-28}-166q^{-30}+112q^{-32}-74q^{-34}+44q^{-36}-22q^{-38}+10q^{-40}-4q^{-42}+q^{-44}$
2,0	$q^{42}-q^{40}-2q^{38}+q^{36}+4q^{34}-8q^{30}-q^{28}+9q^{26}+q^{24}-11q^{22}-q^{20}+14q^{18}+4q^{16}-13q^{14}+12q^{10}-2q^{8}-7q^{6}+2q^{4}+2q^{2}-2+2q^{-2}+2q^{-4}-7q^{-6}-2q^{-8}+12q^{-10}-13q^{-14}+4q^{-16}+14q^{-18}-q^{-20}-11q^{-22}+q^{-24}+9q^{-26}-q^{-28}-8q^{-30}+4q^{-34}+q^{-36}-2q^{-38}-q^{-40}+q^{-42}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-2q^{32}+3q^{28}-6q^{26}+3q^{24}+8q^{22}-11q^{20}+3q^{18}+12q^{16}-16q^{14}-q^{12}+13q^{10}-11q^{8}-3q^{6}+10q^{4}+q^{2}-2+q^{-2}+10q^{-4}-3q^{-6}-11q^{-8}+13q^{-10}-q^{-12}-16q^{-14}+12q^{-16}+3q^{-18}-11q^{-20}+8q^{-22}+3q^{-24}-6q^{-26}+3q^{-28}-2q^{-32}+q^{-34}$
1,0,0	$-q^{21}+q^{19}+q^{15}-2q^{13}+2q^{11}-q^{9}+q^{7}-q^{5}+2q^{3}+2q^{-3}-q^{-5}+q^{-7}-q^{-9}+2q^{-11}-2q^{-13}+q^{-15}+q^{-19}-q^{-21}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+2q^{32}-4q^{30}+7q^{28}-10q^{26}+13q^{24}-16q^{22}+17q^{20}-17q^{18}+16q^{16}-10q^{14}+5q^{12}+5q^{10}-13q^{8}+21q^{6}-28q^{4}+33q^{2}-36+33q^{-2}-28q^{-4}+21q^{-6}-13q^{-8}+5q^{-10}+5q^{-12}-10q^{-14}+16q^{-16}-17q^{-18}+17q^{-20}-16q^{-22}+13q^{-24}-10q^{-26}+7q^{-28}-4q^{-30}+2q^{-32}-q^{-34}$
1,0	$q^{56}-2q^{52}-2q^{50}+2q^{48}+5q^{46}-q^{44}-8q^{42}-4q^{40}+9q^{38}+12q^{36}-4q^{34}-16q^{32}-5q^{30}+15q^{28}+15q^{26}-9q^{24}-19q^{22}-2q^{20}+17q^{18}+8q^{16}-12q^{14}-11q^{12}+7q^{10}+12q^{8}-3q^{6}-11q^{4}+2q^{2}+13+2q^{-2}-11q^{-4}-3q^{-6}+12q^{-8}+7q^{-10}-11q^{-12}-12q^{-14}+8q^{-16}+17q^{-18}-2q^{-20}-19q^{-22}-9q^{-24}+15q^{-26}+15q^{-28}-5q^{-30}-16q^{-32}-4q^{-34}+12q^{-36}+9q^{-38}-4q^{-40}-8q^{-42}-q^{-44}+5q^{-46}+2q^{-48}-2q^{-50}-2q^{-52}+q^{-56}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-2q^{78}+4q^{76}-7q^{74}+6q^{72}-5q^{70}-2q^{68}+14q^{66}-23q^{64}+32q^{62}-32q^{60}+19q^{58}+3q^{56}-33q^{54}+60q^{52}-73q^{50}+67q^{48}-37q^{46}-9q^{44}+57q^{42}-88q^{40}+96q^{38}-70q^{36}+20q^{34}+31q^{32}-69q^{30}+73q^{28}-46q^{26}+45q^{22}-65q^{20}+47q^{18}-3q^{16}-55q^{14}+102q^{12}-111q^{10}+80q^{8}-14q^{6}-61q^{4}+124q^{2}-145+124q^{-2}-61q^{-4}-14q^{-6}+80q^{-8}-111q^{-10}+102q^{-12}-55q^{-14}-3q^{-16}+47q^{-18}-65q^{-20}+45q^{-22}-46q^{-26}+73q^{-28}-69q^{-30}+31q^{-32}+20q^{-34}-70q^{-36}+96q^{-38}-88q^{-40}+57q^{-42}-9q^{-44}-37q^{-46}+67q^{-48}-73q^{-50}+60q^{-52}-33q^{-54}+3q^{-56}+19q^{-58}-32q^{-60}+32q^{-62}-23q^{-64}+14q^{-66}-2q^{-68}-5q^{-70}+6q^{-72}-7q^{-74}+4q^{-76}-2q^{-78}+q^{-80}$

.

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 33"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$4t^{2}-16t+25-16t^{-1}+4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$4z^{4}+1$

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]=

{ 65, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+3z^{8}a^{-2}+6z^{8}+4a^{3}z^{7}+5az^{7}+5z^{7}a^{-1}+4z^{7}a^{-3}+3a^{4}z^{6}-4a^{2}z^{6}-4z^{6}a^{-2}+3z^{6}a^{-4}-14z^{6}+a^{5}z^{5}-9a^{3}z^{5}-16az^{5}-16z^{5}a^{-1}-9z^{5}a^{-3}+z^{5}a^{-5}-7a^{4}z^{4}+a^{2}z^{4}+z^{4}a^{-2}-7z^{4}a^{-4}+16z^{4}-2a^{5}z^{3}+6a^{3}z^{3}+18az^{3}+18z^{3}a^{-1}+6z^{3}a^{-3}-2z^{3}a^{-5}+3a^{4}z^{2}+3z^{2}a^{-4}-6z^{2}-2a^{3}z-6az-6za^{-1}-2za^{-3}+1$

Vassiliev invariants

V2 and V3:

(0, 0)

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 $0$ $0$ $0$ $-32$ $-32$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $-16$ $-{\frac {64}{3}}$ ${\frac {128}{3}}$ $-{\frac {112}{3}}$ $-16$

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=$0 is the signature of 10 33. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	-5	-4	-3	-2	-1	0	1	2	3	4	5	χ
11											1	-1
9										2		2
7									3	1		-2
5								5	2			3
3							5	3				-2
1						6	5					1
-1					5	6						1
-3				3	5							-2
-5			2	5								3
-7		1	3									-2
-9		2										2
-11	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, 33]]
Out[2]=	10
In[3]:=	PD[Knot[10, 33]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 6, 15, 5], X[20, 15, 1, 16], X[16, 7, 17, 8], X[8, 19, 9, 20], X[18, 9, 19, 10], X[10, 17, 11, 18], X[2, 14, 3, 13], X[12, 4, 13, 3], X[4, 12, 5, 11]]
In[4]:=	GaussCode[Knot[10, 33]]
Out[4]=	GaussCode[1, -8, 9, -10, 2, -1, 4, -5, 6, -7, 10, -9, 8, -2, 3, -4, 7, -6, 5, -3]
In[5]:=	BR[Knot[10, 33]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, 3, -2, 3, 3, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[10, 33]][t]
Out[6]=	4 16 2 25 + -- - -- - 16 t + 4 t 2 t t
In[7]:=	Conway[Knot[10, 33]][z]
Out[7]=	4 1 + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 33], Knot[11, Alternating, 333]}
In[9]:=	{KnotDet[Knot[10, 33]], KnotSignature[Knot[10, 33]]}
Out[9]=	{65, 0}
In[10]:=	J=Jones[Knot[10, 33]][q]
Out[10]=	-5 3 5 8 10 2 3 4 5 11 - q + -- - -- + -- - -- - 10 q + 8 q - 5 q + 3 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 33]}
In[12]:=	A2Invariant[Knot[10, 33]][q]
Out[12]=	-16 -14 -12 2 2 -4 2 2 4 8 -1 - q + q + q - --- + -- - q + -- + 2 q - q + 2 q - 10 8 2 q q q 10 12 14 16 2 q + q + q - q
In[13]:=	Kauffman[Knot[10, 33]][a, z]
Out[13]=	2 3 3 2 z 6 z 3 2 3 z 4 2 2 z 6 z 1 - --- - --- - 6 a z - 2 a z - 6 z + ---- + 3 a z - ---- + ---- + 3 a 4 5 3 a a a a 3 4 4 18 z 3 3 3 5 3 4 7 z z 2 4 ----- + 18 a z + 6 a z - 2 a z + 16 z - ---- + -- + a z - a 4 2 a a 5 5 5 4 4 z 9 z 16 z 5 3 5 5 5 6 7 a z + -- - ---- - ----- - 16 a z - 9 a z + a z - 14 z + 5 3 a a a 6 6 7 7 3 z 4 z 2 6 4 6 4 z 5 z 7 3 7 ---- - ---- - 4 a z + 3 a z + ---- + ---- + 5 a z + 4 a z + 4 2 3 a a a a 8 9 8 3 z 2 8 z 9 6 z + ---- + 3 a z + -- + a z 2 a a
In[14]:=	{Vassiliev[2][Knot[10, 33]], Vassiliev[3][Knot[10, 33]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 33]][q, t]
Out[15]=	6 1 2 1 3 2 5 3 - + 6 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 5 5 3 3 2 5 2 5 3 7 3 ---- + --- + 5 q t + 5 q t + 3 q t + 5 q t + 2 q t + 3 q t + 3 q t q t 7 4 9 4 11 5 q t + 2 q t + q t