9 14: Difference between revisions

Revision as of 20:15, 28 August 2005

9_13

9_15

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9 14 Quick Notes

9 14 Further Notes and Views

Knot presentations

Planar diagram presentation	X₁₄₂₅ X_5,12,6,13 X_3,11,4,10 X_11,3,12,2 X_13,18,14,1 X_9,15,10,14 X_7,17,8,16 X_15,9,16,8 X_17,7,18,6
Gauss code	-1, 4, -3, 1, -2, 9, -7, 8, -6, 3, -4, 2, -5, 6, -8, 7, -9, 5
Dowker-Thistlethwaite code	4 10 12 16 14 2 18 8 6
Conway Notation	[41112]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	1
3-genus	2
Bridge index	2
Super bridge index	$\{4,7\}$
Nakanishi index	1
Maximal Thurston-Bennequin number	[-4][-7]
Hyperbolic Volume	8.95499
A-Polynomial	See Data:9 14/A-polynomial

[edit Notes for 9 14'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 9 14's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{2}-9t+15-9t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 37, 0 }
Jones polynomial	$q^{6}-2q^{5}+3q^{4}-5q^{3}+6q^{2}-6q+6-4q^{-1}+3q^{-2}-q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-2}+z^{4}-a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}+a^{-2}-2a^{-4}+a^{-6}+1$
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}a^{-4}+3z^{7}a^{-1}+5z^{7}a^{-3}+2z^{7}a^{-5}+3z^{6}a^{-2}+z^{6}a^{-6}+4z^{6}+4az^{5}-4z^{5}a^{-1}-16z^{5}a^{-3}-8z^{5}a^{-5}+3a^{2}z^{4}-12z^{4}a^{-2}-9z^{4}a^{-4}-4z^{4}a^{-6}-4z^{4}+a^{3}z^{3}-3az^{3}+2z^{3}a^{-1}+15z^{3}a^{-3}+9z^{3}a^{-5}-2a^{2}z^{2}+8z^{2}a^{-2}+10z^{2}a^{-4}+4z^{2}a^{-6}-2za^{-1}-5za^{-3}-3za^{-5}-a^{-2}-2a^{-4}-a^{-6}+1$
The A2 invariant	$-q^{10}+q^{8}+q^{6}-q^{4}+2q^{2}+q^{-2}+q^{-4}+q^{-8}-2q^{-10}-q^{-12}-q^{-16}+q^{-18}+q^{-20}$
The G2 invariant	$q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{44}-3q^{40}+8q^{38}-10q^{36}+11q^{34}-8q^{32}+2q^{30}+4q^{28}-10q^{26}+16q^{24}-16q^{22}+14q^{20}-8q^{18}-q^{16}+11q^{14}-15q^{12}+17q^{10}-12q^{8}+3q^{6}+6q^{4}-10q^{2}+10-2q^{-2}-6q^{-4}+15q^{-6}-16q^{-8}+9q^{-10}+5q^{-12}-19q^{-14}+30q^{-16}-28q^{-18}+18q^{-20}-16q^{-24}+28q^{-26}-30q^{-28}+23q^{-30}-10q^{-32}-6q^{-34}+16q^{-36}-19q^{-38}+15q^{-40}-5q^{-42}-7q^{-44}+11q^{-46}-13q^{-48}+5q^{-50}+5q^{-52}-16q^{-54}+21q^{-56}-19q^{-58}+6q^{-60}+8q^{-62}-20q^{-64}+25q^{-66}-21q^{-68}+11q^{-70}+q^{-72}-10q^{-74}+16q^{-76}-14q^{-78}+10q^{-80}-2q^{-82}-2q^{-84}+3q^{-86}-4q^{-88}+3q^{-90}-q^{-92}+q^{-94}$

Further Quantum Invariants

Further quantum knot invariants for 9_14.

A1 Invariants.

Weight	Invariant
1	$-q^{7}+2q^{5}-q^{3}+2q+q^{-5}-2q^{-7}+q^{-9}-q^{-11}+q^{-13}$
2	$q^{20}-2q^{18}-q^{16}+4q^{14}-4q^{12}+7q^{8}-5q^{6}-q^{4}+7q^{2}-3-3q^{-2}+3q^{-4}+2q^{-6}-3q^{-8}-2q^{-10}+5q^{-12}-q^{-14}-6q^{-16}+5q^{-18}+2q^{-20}-6q^{-22}+4q^{-24}+4q^{-26}-5q^{-28}+3q^{-32}-2q^{-34}-q^{-36}+q^{-38}$
3	$-q^{39}+2q^{37}+q^{35}-2q^{33}-2q^{31}+q^{29}+3q^{27}-5q^{25}+6q^{21}-10q^{17}+3q^{15}+14q^{13}-q^{11}-17q^{9}+19q^{5}+5q^{3}-16q-9q^{-1}+10q^{-3}+11q^{-5}-2q^{-7}-14q^{-9}-3q^{-11}+12q^{-13}+10q^{-15}-11q^{-17}-13q^{-19}+8q^{-21}+16q^{-23}-6q^{-25}-16q^{-27}+2q^{-29}+18q^{-31}+2q^{-33}-17q^{-35}-6q^{-37}+16q^{-39}+12q^{-41}-13q^{-43}-15q^{-45}+5q^{-47}+16q^{-49}-q^{-51}-14q^{-53}-4q^{-55}+10q^{-57}+7q^{-59}-5q^{-61}-6q^{-63}+2q^{-65}+4q^{-67}-2q^{-71}-q^{-73}+q^{-75}$
4	$q^{64}-2q^{62}-q^{60}+2q^{58}+5q^{54}-4q^{52}-2q^{50}-6q^{46}+11q^{44}-2q^{42}+3q^{40}-21q^{36}+4q^{34}+6q^{32}+26q^{30}+10q^{28}-43q^{26}-22q^{24}+4q^{22}+56q^{20}+40q^{18}-48q^{16}-56q^{14}-24q^{12}+60q^{10}+70q^{8}-16q^{6}-54q^{4}-55q^{2}+21+63q^{-2}+24q^{-4}-14q^{-6}-52q^{-8}-26q^{-10}+19q^{-12}+41q^{-14}+30q^{-16}-25q^{-18}-48q^{-20}-17q^{-22}+41q^{-24}+47q^{-26}-3q^{-28}-53q^{-30}-36q^{-32}+39q^{-34}+51q^{-36}+8q^{-38}-57q^{-40}-48q^{-42}+34q^{-44}+54q^{-46}+29q^{-48}-47q^{-50}-64q^{-52}+8q^{-54}+45q^{-56}+56q^{-58}-14q^{-60}-63q^{-62}-29q^{-64}+8q^{-66}+62q^{-68}+30q^{-70}-28q^{-72}-41q^{-74}-34q^{-76}+32q^{-78}+44q^{-80}+15q^{-82}-17q^{-84}-47q^{-86}-6q^{-88}+21q^{-90}+27q^{-92}+12q^{-94}-25q^{-96}-17q^{-98}-4q^{-100}+12q^{-102}+16q^{-104}-4q^{-106}-6q^{-108}-6q^{-110}+6q^{-114}+q^{-116}-2q^{-120}-q^{-122}+q^{-124}$
5	$-q^{95}+2q^{93}+q^{91}-2q^{89}-3q^{85}-2q^{83}+3q^{81}+7q^{79}+3q^{77}-q^{75}-8q^{73}-12q^{71}-4q^{69}+13q^{67}+26q^{65}+10q^{63}-15q^{61}-36q^{59}-38q^{57}+9q^{55}+66q^{53}+62q^{51}+q^{49}-80q^{47}-112q^{45}-33q^{43}+111q^{41}+167q^{39}+71q^{37}-114q^{35}-226q^{33}-137q^{31}+104q^{29}+277q^{27}+209q^{25}-60q^{23}-298q^{21}-276q^{19}-8q^{17}+276q^{15}+324q^{13}+90q^{11}-216q^{9}-326q^{7}-162q^{5}+119q^{3}+282q+208q^{-1}-17q^{-3}-202q^{-5}-216q^{-7}-66q^{-9}+103q^{-11}+184q^{-13}+132q^{-15}-13q^{-17}-142q^{-19}-158q^{-21}-53q^{-23}+90q^{-25}+172q^{-27}+104q^{-29}-65q^{-31}-169q^{-33}-121q^{-35}+46q^{-37}+177q^{-39}+134q^{-41}-48q^{-43}-189q^{-45}-148q^{-47}+46q^{-49}+209q^{-51}+168q^{-53}-43q^{-55}-223q^{-57}-200q^{-59}+18q^{-61}+233q^{-63}+234q^{-65}+20q^{-67}-213q^{-69}-260q^{-71}-80q^{-73}+172q^{-75}+272q^{-77}+137q^{-79}-105q^{-81}-250q^{-83}-190q^{-85}+19q^{-87}+202q^{-89}+217q^{-91}+59q^{-93}-122q^{-95}-199q^{-97}-131q^{-99}+32q^{-101}+157q^{-103}+159q^{-105}+50q^{-107}-79q^{-109}-149q^{-111}-112q^{-113}+3q^{-115}+104q^{-117}+127q^{-119}+61q^{-121}-42q^{-123}-107q^{-125}-95q^{-127}-16q^{-129}+65q^{-131}+93q^{-133}+51q^{-135}-18q^{-137}-65q^{-139}-62q^{-141}-15q^{-143}+34q^{-145}+51q^{-147}+27q^{-149}-7q^{-151}-29q^{-153}-28q^{-155}-6q^{-157}+14q^{-159}+17q^{-161}+7q^{-163}-2q^{-165}-8q^{-167}-8q^{-169}+4q^{-173}+3q^{-175}+q^{-177}-2q^{-181}-q^{-183}+q^{-185}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{10}+q^{8}+q^{6}-q^{4}+2q^{2}+q^{-2}+q^{-4}+q^{-8}-2q^{-10}-q^{-12}-q^{-16}+q^{-18}+q^{-20}$
1,1	$q^{28}-4q^{26}+8q^{24}-12q^{22}+18q^{20}-28q^{18}+34q^{16}-38q^{14}+43q^{12}-48q^{10}+50q^{8}-44q^{6}+46q^{4}-32q^{2}+22-24q^{-4}+50q^{-6}-80q^{-8}+102q^{-10}-118q^{-12}+126q^{-14}-126q^{-16}+108q^{-18}-91q^{-20}+62q^{-22}-30q^{-24}+34q^{-28}-50q^{-30}+70q^{-32}-76q^{-34}+71q^{-36}-62q^{-38}+48q^{-40}-36q^{-42}+21q^{-44}-12q^{-46}+6q^{-48}-2q^{-50}+q^{-52}$
2,0	$q^{26}-q^{24}-2q^{22}+q^{20}+2q^{18}-q^{16}-4q^{14}+3q^{12}+5q^{10}-3q^{8}-2q^{6}+6q^{4}+4q^{2}-2-2q^{-2}+2q^{-4}-q^{-8}+q^{-10}-3q^{-14}+2q^{-16}+q^{-18}-5q^{-20}-3q^{-22}+2q^{-24}+3q^{-26}-2q^{-28}+5q^{-32}+4q^{-34}-2q^{-36}-2q^{-38}+q^{-40}+q^{-42}-2q^{-44}-3q^{-46}+q^{-50}+q^{-52}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{22}-2q^{20}-q^{18}+4q^{16}-3q^{14}-2q^{12}+6q^{10}-2q^{8}-4q^{6}+7q^{4}+q^{2}-1+5q^{-2}+2q^{-4}-2q^{-6}-3q^{-8}-6q^{-14}+2q^{-16}+5q^{-18}-4q^{-20}+2q^{-22}+4q^{-24}-4q^{-26}+2q^{-30}-3q^{-32}+q^{-34}+q^{-36}-q^{-38}+q^{-40}$
1,0,0	$-q^{13}+q^{11}+q^{7}-q^{5}+2q^{3}+q^{-1}+q^{-3}+q^{-5}+q^{-7}+q^{-11}-2q^{-13}-q^{-15}-2q^{-17}-q^{-21}+q^{-23}+q^{-25}+q^{-27}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{22}+2q^{20}-3q^{18}+4q^{16}-5q^{14}+6q^{12}-6q^{10}+6q^{8}-4q^{6}+3q^{4}+q^{2}-3+7q^{-2}-8q^{-4}+12q^{-6}-11q^{-8}+12q^{-10}-10q^{-12}+8q^{-14}-6q^{-16}+q^{-18}-4q^{-22}+4q^{-24}-6q^{-26}+6q^{-28}-6q^{-30}+5q^{-32}-3q^{-34}+3q^{-36}-q^{-38}+q^{-40}$
1,0	$q^{36}-2q^{32}-2q^{30}+q^{28}+4q^{26}+q^{24}-4q^{22}-4q^{20}+2q^{18}+6q^{16}+2q^{14}-5q^{12}-4q^{10}+3q^{8}+7q^{6}-4q^{2}+6q^{-2}+3q^{-4}-3q^{-6}-4q^{-8}+2q^{-10}+3q^{-12}-2q^{-14}-5q^{-16}+3q^{-20}-q^{-22}-5q^{-24}-q^{-26}+6q^{-28}+4q^{-30}-3q^{-32}-6q^{-34}+2q^{-36}+7q^{-38}+3q^{-40}-5q^{-42}-5q^{-44}+2q^{-46}+5q^{-48}-4q^{-52}-2q^{-54}+2q^{-56}+2q^{-58}-q^{-60}-q^{-62}+q^{-66}$

G2 Invariants.

Weight

Invariant

1,0

q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{44}-3q^{40}+8q^{38}-10q^{36}+11q^{34}-8q^{32}+2q^{30}+4q^{28}-10q^{26}+16q^{24}-16q^{22}+14q^{20}-8q^{18}-q^{16}+11q^{14}-15q^{12}+17q^{10}-12q^{8}+3q^{6}+6q^{4}-10q^{2}+10-2q^{-2}-6q^{-4}+15q^{-6}-16q^{-8}+9q^{-10}+5q^{-12}-19q^{-14}+30q^{-16}-28q^{-18}+18q^{-20}-16q^{-24}+28q^{-26}-30q^{-28}+23q^{-30}-10q^{-32}-6q^{-34}+16q^{-36}-19q^{-38}+15q^{-40}-5q^{-42}-7q^{-44}+11q^{-46}-13q^{-48}+5q^{-50}+5q^{-52}-16q^{-54}+21q^{-56}-19q^{-58}+6q^{-60}+8q^{-62}-20q^{-64}+25q^{-66}-21q^{-68}+11q^{-70}+q^{-72}-10q^{-74}+16q^{-76}-14q^{-78}+10q^{-80}-2q^{-82}-2q^{-84}+3q^{-86}-4q^{-88}+3q^{-90}-q^{-92}+q^{-94}

.

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 14"];

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

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

Out[4]= $2t^{2}-9t+15-9t^{-1}+2t^{-2}$

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

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

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, 0 }

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

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

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

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

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

Out[9]= $z^{4}a^{-2}+z^{4}-a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}+a^{-2}-2a^{-4}+a^{-6}+1$

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

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

Out[10]= $z^{8}a^{-2}+z^{8}a^{-4}+3z^{7}a^{-1}+5z^{7}a^{-3}+2z^{7}a^{-5}+3z^{6}a^{-2}+z^{6}a^{-6}+4z^{6}+4az^{5}-4z^{5}a^{-1}-16z^{5}a^{-3}-8z^{5}a^{-5}+3a^{2}z^{4}-12z^{4}a^{-2}-9z^{4}a^{-4}-4z^{4}a^{-6}-4z^{4}+a^{3}z^{3}-3az^{3}+2z^{3}a^{-1}+15z^{3}a^{-3}+9z^{3}a^{-5}-2a^{2}z^{2}+8z^{2}a^{-2}+10z^{2}a^{-4}+4z^{2}a^{-6}-2za^{-1}-5za^{-3}-3za^{-5}-a^{-2}-2a^{-4}-a^{-6}+1$

Vassiliev invariants

V₂ and V₃:

(-1, -2)

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

-4

-16

8

-{\frac {110}{3}}

-{\frac {34}{3}}

64

{\frac {32}{3}}

{\frac {128}{3}}

-48

-{\frac {32}{3}}

128

{\frac {440}{3}}

{\frac {136}{3}}

{\frac {10529}{30}}

{\frac {1502}{15}}

{\frac {4978}{45}}

-{\frac {833}{18}}

{\frac {449}{30}}

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 9 14. 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

χ

13

1

11

1

-1

9

2

1

7

3

1

-2

5

3

2

1

3

0

1

3

0

-1

2

4

2

-3

1

2

-1

-5

2

-7

1

-1

Integral Khovanov Homology

(db, data source)

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

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[9, 14]]
Out[2]=	9
In[3]:=	PD[Knot[9, 14]]
Out[3]=	PD[X[1, 4, 2, 5], X[5, 12, 6, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[13, 18, 14, 1], X[9, 15, 10, 14], X[7, 17, 8, 16], X[15, 9, 16, 8], X[17, 7, 18, 6]]
In[4]:=	GaussCode[Knot[9, 14]]
Out[4]=	GaussCode[-1, 4, -3, 1, -2, 9, -7, 8, -6, 3, -4, 2, -5, 6, -8, 7, -9, 5]
In[5]:=	BR[Knot[9, 14]]
Out[5]=	BR[5, {1, 1, 2, -1, -3, 2, -3, 4, -3, 4}]
In[6]:=	alex = Alexander[Knot[9, 14]][t]
Out[6]=	2 9 2 15 + -- - - - 9 t + 2 t 2 t t
In[7]:=	Conway[Knot[9, 14]][z]
Out[7]=	2 4 1 - z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 14]}
In[9]:=	{KnotDet[Knot[9, 14]], KnotSignature[Knot[9, 14]]}
Out[9]=	{37, 0}
In[10]:=	J=Jones[Knot[9, 14]][q]
Out[10]=	-3 3 4 2 3 4 5 6 6 - q + -- - - - 6 q + 6 q - 5 q + 3 q - 2 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 14], Knot[11, NonAlternating, 53]}
In[12]:=	A2Invariant[Knot[9, 14]][q]
Out[12]=	-10 -8 -6 -4 2 2 4 8 10 12 16 18 -q + q + q - q + -- + q + q + q - 2 q - q - q + q + 2 q 20 q
In[13]:=	Kauffman[Knot[9, 14]][a, z]
Out[13]=	2 2 2 -6 2 -2 3 z 5 z 2 z 4 z 10 z 8 z 2 2 1 - a - -- - a - --- - --- - --- + ---- + ----- + ---- - 2 a z + 4 5 3 a 6 4 2 a a a a a a 3 3 3 4 4 4 9 z 15 z 2 z 3 3 3 4 4 z 9 z 12 z ---- + ----- + ---- - 3 a z + a z - 4 z - ---- - ---- - ----- + 5 3 a 6 4 2 a a a a a 5 5 5 6 6 7 2 4 8 z 16 z 4 z 5 6 z 3 z 2 z 3 a z - ---- - ----- - ---- + 4 a z + 4 z + -- + ---- + ---- + 5 3 a 6 2 5 a a a a a 7 7 8 8 5 z 3 z z z ---- + ---- + -- + -- 3 a 4 2 a a a
In[14]:=	{Vassiliev[2][Knot[9, 14]], Vassiliev[3][Knot[9, 14]]}
Out[14]=	{0, -2}
In[15]:=	Kh[Knot[9, 14]][q, t]
Out[15]=	4 1 2 1 2 2 3 - + 3 q + ----- + ----- + ----- + ---- + --- + 3 q t + 3 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 3 2 5 2 5 3 7 3 7 4 9 4 9 5 3 q t + 3 q t + 2 q t + 3 q t + q t + 2 q t + q t + 11 5 13 6 q t + q t

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-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>.
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 <tr align=center><td>-7</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>}}
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   q   t  + q   t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

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Revision as of 20:15, 28 August 2005

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Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

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