K11a297

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a297 at Knotilus!
	[edit K11a297 Quick Notes]

[edit K11a297 Further Notes and Views]

Knot presentations

Planar diagram presentation	X₆₂₇₁ X_10,3,11,4 X_16,5,17,6 X_22,8,1,7 X_20,9,21,10 X_18,12,19,11 X_2,13,3,14 X_8,15,9,16 X_12,18,13,17 X_4,19,5,20 X_14,21,15,22
Gauss code	1, -7, 2, -10, 3, -1, 4, -8, 5, -2, 6, -9, 7, -11, 8, -3, 9, -6, 10, -5, 11, -4
Dowker-Thistlethwaite code	6 10 16 22 20 18 2 8 12 4 14

A Braid Representative

A Morse Link Presentation

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	$\{1,2\}$
3-genus	3
Bridge index	3
Super bridge index	Missing
Nakanishi index	Missing
Maximal Thurston-Bennequin number	Data:K11a297/ThurstonBennequinNumber
Hyperbolic Volume	18.8513
A-Polynomial	See Data:K11a297/A-polynomial

[edit Notes for K11a297's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	Missing
Topological 4 genus	Missing
Concordance genus	$[1,3]$
Rasmussen s-Invariant	2

[edit Notes for K11a297's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-15t^{2}+42t-57+42t^{-1}-15t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}-3z^{4}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{2}-3t+1\right\}$
Determinant and Signature	{ 175, -2 }
Jones polynomial	$q^{3}-4q^{2}+10q-17+24q^{-1}-28q^{-2}+29q^{-3}-25q^{-4}+19q^{-5}-12q^{-6}+5q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{6}-a^{6}+z^{6}a^{4}+z^{4}a^{4}+3z^{2}a^{4}+3a^{4}+z^{6}a^{2}-z^{4}a^{2}-4z^{2}a^{2}-3a^{2}-2z^{4}+2+z^{2}a^{-2}$
Kauffman polynomial (db, data sources)	$4a^{4}z^{10}+4a^{2}z^{10}+12a^{5}z^{9}+21a^{3}z^{9}+9az^{9}+16a^{6}z^{8}+19a^{4}z^{8}+11a^{2}z^{8}+8z^{8}+12a^{7}z^{7}-9a^{5}z^{7}-41a^{3}z^{7}-16az^{7}+4z^{7}a^{-1}+5a^{8}z^{6}-24a^{6}z^{6}-51a^{4}z^{6}-40a^{2}z^{6}+z^{6}a^{-2}-17z^{6}+a^{9}z^{5}-15a^{7}z^{5}-7a^{5}z^{5}+27a^{3}z^{5}+10az^{5}-8z^{5}a^{-1}-3a^{8}z^{4}+11a^{6}z^{4}+37a^{4}z^{4}+38a^{2}z^{4}-2z^{4}a^{-2}+13z^{4}+4a^{7}z^{3}+a^{5}z^{3}-12a^{3}z^{3}-5az^{3}+4z^{3}a^{-1}-3a^{6}z^{2}-13a^{4}z^{2}-17a^{2}z^{2}+z^{2}a^{-2}-6z^{2}+2a^{5}z+4a^{3}z+2az+a^{6}+3a^{4}+3a^{2}+2$
The A2 invariant	Data:K11a297/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a297/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a297 Quantum Invariants.

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["K11a297"];

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

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

Out[4]= $2t^{3}-15t^{2}+42t-57+42t^{-1}-15t^{-2}+2t^{-3}$

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

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

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]= $\left\{t^{2}-3t+1\right\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 175, -2 }

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

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

Out[8]= $q^{3}-4q^{2}+10q-17+24q^{-1}-28q^{-2}+29q^{-3}-25q^{-4}+19q^{-5}-12q^{-6}+5q^{-7}-q^{-8}$

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

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

Out[9]= $-z^{4}a^{6}-a^{6}+z^{6}a^{4}+z^{4}a^{4}+3z^{2}a^{4}+3a^{4}+z^{6}a^{2}-z^{4}a^{2}-4z^{2}a^{2}-3a^{2}-2z^{4}+2+z^{2}a^{-2}$

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

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

Out[10]= $4a^{4}z^{10}+4a^{2}z^{10}+12a^{5}z^{9}+21a^{3}z^{9}+9az^{9}+16a^{6}z^{8}+19a^{4}z^{8}+11a^{2}z^{8}+8z^{8}+12a^{7}z^{7}-9a^{5}z^{7}-41a^{3}z^{7}-16az^{7}+4z^{7}a^{-1}+5a^{8}z^{6}-24a^{6}z^{6}-51a^{4}z^{6}-40a^{2}z^{6}+z^{6}a^{-2}-17z^{6}+a^{9}z^{5}-15a^{7}z^{5}-7a^{5}z^{5}+27a^{3}z^{5}+10az^{5}-8z^{5}a^{-1}-3a^{8}z^{4}+11a^{6}z^{4}+37a^{4}z^{4}+38a^{2}z^{4}-2z^{4}a^{-2}+13z^{4}+4a^{7}z^{3}+a^{5}z^{3}-12a^{3}z^{3}-5az^{3}+4z^{3}a^{-1}-3a^{6}z^{2}-13a^{4}z^{2}-17a^{2}z^{2}+z^{2}a^{-2}-6z^{2}+2a^{5}z+4a^{3}z+2az+a^{6}+3a^{4}+3a^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$ ): {K11a125,}

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.

In[3]:= K = Knot["K11a297"];

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]= { $2t^{3}-15t^{2}+42t-57+42t^{-1}-15t^{-2}+2t^{-3}$ , $q^{3}-4q^{2}+10q-17+24q^{-1}-28q^{-2}+29q^{-3}-25q^{-4}+19q^{-5}-12q^{-6}+5q^{-7}-q^{-8}$ }

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]= {}

In[6]:= DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

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

Out[6]= {K11a125,}

Vassiliev invariants

V₂ and V₃:

(0, -1)

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
$0$	$-8$	$0$	$48$	$24$	$0$	$-{\frac {368}{3}}$	${\frac {64}{3}}$	$-104$	$0$	$32$	$0$	$0$	$264$	$-392$	$520$	$56$	$104$

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 K11a297. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

χ

7

1

5

3

-3

3

7

1

6

1

10

3

-7

-1

14

7

-3

15

11

-4

-5

14

13

1

-7

11

15

4

-9

8

14

-6

-11

4

11

7

-13

1

8

-7

-15

4

-17

1

-1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=-3$	$i=-1$
$r=-7$	${\mathbb {Z} }$
$r=-6$	${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=-5$	${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{4}$	${\mathbb {Z} }^{4}$
$r=-4$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{8}$	${\mathbb {Z} }^{8}$
$r=-3$	${\mathbb {Z} }^{14}\oplus {\mathbb {Z} }_{2}^{11}$	${\mathbb {Z} }^{11}$
$r=-2$	${\mathbb {Z} }^{15}\oplus {\mathbb {Z} }_{2}^{14}$	${\mathbb {Z} }^{14}$
$r=-1$	${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{15}$	${\mathbb {Z} }^{15}$
$r=0$	${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{13}$	${\mathbb {Z} }^{14}$
$r=1$	${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{10}$	${\mathbb {Z} }^{10}$
$r=2$	${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$	${\mathbb {Z} }^{7}$
$r=3$	${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$	${\mathbb {Z} }^{3}$
$r=4$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$