K11n97

K11n96

K11n98

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n97 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X14,6,15,5 X12,8,13,7 X9,19,10,18 X2,11,3,12 X6,14,7,13 X15,20,16,21 X17,22,18,1 X19,9,20,8 X21,16,22,17
Gauss code	1, -6, 2, -1, 3, -7, 4, 10, -5, -2, 6, -4, 7, -3, -8, 11, -9, 5, -10, 8, -11, 9
Dowker-Thistlethwaite code	4 10 14 12 -18 2 6 -20 -22 -8 -16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n97/ThurstonBennequinNumber Hyperbolic Volume 10.183 A-Polynomial See Data:K11n97/A-polynomial

[edit Notes for K11n97's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $0$ Rasmussen s-Invariant 0

[edit Notes for K11n97's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-2t+5-2t^{-1}$
Conway polynomial	$1-2z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 9, 0 }
Jones polynomial	$q^{5}-q^{4}+q^{3}-q^{2}-q+2-2q^{-1}+3q^{-2}-2q^{-3}+2q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{4}-a^{4}+z^{4}a^{2}+3z^{2}a^{2}+3a^{2}-z^{2}-z^{4}a^{-2}-4z^{2}a^{-2}-3a^{-2}+z^{2}a^{-4}+2a^{-4}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-1}+z^{9}a^{-3}+2z^{8}a^{-2}+z^{8}a^{-4}+z^{8}+a^{3}z^{7}-8z^{7}a^{-1}-7z^{7}a^{-3}+2a^{4}z^{6}+2a^{2}z^{6}-15z^{6}a^{-2}-7z^{6}a^{-4}-8z^{6}+a^{5}z^{5}-2a^{3}z^{5}+az^{5}+19z^{5}a^{-1}+15z^{5}a^{-3}-7a^{4}z^{4}-7a^{2}z^{4}+32z^{4}a^{-2}+15z^{4}a^{-4}+17z^{4}-3a^{5}z^{3}-2a^{3}z^{3}-4az^{3}-18z^{3}a^{-1}-13z^{3}a^{-3}+5a^{4}z^{2}+7a^{2}z^{2}-22z^{2}a^{-2}-11z^{2}a^{-4}-9z^{2}+a^{5}z+2a^{3}z+4az+8za^{-1}+5za^{-3}-a^{4}-3a^{2}+3a^{-2}+2a^{-4}$
The A2 invariant	Data:K11n97/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n97/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n97 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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11n97"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-2t+5-2t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$1-2z^{2}$

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]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-1}+z^{9}a^{-3}+2z^{8}a^{-2}+z^{8}a^{-4}+z^{8}+a^{3}z^{7}-8z^{7}a^{-1}-7z^{7}a^{-3}+2a^{4}z^{6}+2a^{2}z^{6}-15z^{6}a^{-2}-7z^{6}a^{-4}-8z^{6}+a^{5}z^{5}-2a^{3}z^{5}+az^{5}+19z^{5}a^{-1}+15z^{5}a^{-3}-7a^{4}z^{4}-7a^{2}z^{4}+32z^{4}a^{-2}+15z^{4}a^{-4}+17z^{4}-3a^{5}z^{3}-2a^{3}z^{3}-4az^{3}-18z^{3}a^{-1}-13z^{3}a^{-3}+5a^{4}z^{2}+7a^{2}z^{2}-22z^{2}a^{-2}-11z^{2}a^{-4}-9z^{2}+a^{5}z+2a^{3}z+4az+8za^{-1}+5za^{-3}-a^{4}-3a^{2}+3a^{-2}+2a^{-4}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {6_1, 9_46, K11n67, K11n139,}

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

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["K11n97"];

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]=

{ $-2t+5-2t^{-1}$, $q^{5}-q^{4}+q^{3}-q^{2}-q+2-2q^{-1}+3q^{-2}-2q^{-3}+2q^{-4}-q^{-5}$ }

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]=

{6_1, 9_46, K11n67, K11n139,}

In[6]:=

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

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

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(-2, -3)

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 $-8$ $-24$ $32$ ${\frac {116}{3}}$ ${\frac {52}{3}}$ $192$ $336$ $96$ $72$ $-{\frac {256}{3}}$ $288$ $-{\frac {928}{3}}$ $-{\frac {416}{3}}$ ${\frac {4409}{15}}$ $-{\frac {796}{15}}$ ${\frac {2876}{45}}$ ${\frac {775}{9}}$ $-{\frac {151}{15}}$

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=$0 is the signature of K11n97. 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 6 χ 11                       1 1 9                         0 7                   1 1   0 5               1 1       0 3             1   1       -2 1           2 2 1         1 -1         2 2             0 -3       1 1 1             1 -5     1 2                 1 -7   1 1                   0 -9   1                     1 -11 1                       -1

Integral Khovanov Homology (db, data source)

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

Computer Talk

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

Modifying This Page

Read me first: Modifying Knot Pages. See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate). See/edit the Hoste-Thistlethwaite_Splice_Base (expert). Back to the top.

K11n96 K11n98