K11a98

K11a97

K11a99

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a98 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X18,7,19,8 X16,9,17,10 X2,11,3,12 X22,14,1,13 X20,16,21,15 X8,17,9,18 X6,19,7,20 X14,22,15,21
Gauss code	1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -11, 8, -5, 9, -4, 10, -8, 11, -7
Dowker-Thistlethwaite code	4 10 12 18 16 2 22 20 8 6 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a98/ThurstonBennequinNumber Hyperbolic Volume 12.0217 A-Polynomial See Data:K11a98/A-polynomial

[edit Notes for K11a98's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a98's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$4t^{2}-19t+31-19t^{-1}+4t^{-2}$
Conway polynomial	$4z^{4}-3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 77, 0 }
Jones polynomial	$-q^{5}+3q^{4}-5q^{3}+9q^{2}-11q+12-12q^{-1}+10q^{-2}-7q^{-3}+4q^{-4}-2q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-2z^{2}a^{4}-a^{4}+z^{4}a^{2}-z^{2}a^{2}+2z^{4}+z^{2}+z^{4}a^{-2}+a^{-2}-z^{2}a^{-4}$
Kauffman polynomial (db, data sources)	$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+5az^{9}+3z^{9}a^{-1}+2a^{4}z^{8}+4z^{8}a^{-2}+2z^{8}+2a^{5}z^{7}-5a^{3}z^{7}-18az^{7}-7z^{7}a^{-1}+4z^{7}a^{-3}+a^{6}z^{6}-4a^{4}z^{6}-4a^{2}z^{6}-8z^{6}a^{-2}+3z^{6}a^{-4}-10z^{6}-7a^{5}z^{5}+6a^{3}z^{5}+33az^{5}+11z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-4a^{6}z^{4}-2a^{4}z^{4}+8a^{2}z^{4}+5z^{4}a^{-2}-7z^{4}a^{-4}+18z^{4}+6a^{5}z^{3}-7a^{3}z^{3}-25az^{3}-7z^{3}a^{-1}+3z^{3}a^{-3}-2z^{3}a^{-5}+4a^{6}z^{2}+4a^{4}z^{2}-6a^{2}z^{2}+z^{2}a^{-2}+3z^{2}a^{-4}-8z^{2}-a^{5}z+3a^{3}z+6az+2za^{-1}-a^{6}-a^{4}-a^{-2}$
The A2 invariant	Data:K11a98/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a98/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a98 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["K11a98"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$4t^{2}-19t+31-19t^{-1}+4t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$4z^{4}-3z^{2}+1$

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]=

{ 77, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+3q^{4}-5q^{3}+9q^{2}-11q+12-12q^{-1}+10q^{-2}-7q^{-3}+4q^{-4}-2q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$a^{6}-2z^{2}a^{4}-a^{4}+z^{4}a^{2}-z^{2}a^{2}+2z^{4}+z^{2}+z^{4}a^{-2}+a^{-2}-z^{2}a^{-4}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{10}+z^{10}+2a^{3}z^{9}+5az^{9}+3z^{9}a^{-1}+2a^{4}z^{8}+4z^{8}a^{-2}+2z^{8}+2a^{5}z^{7}-5a^{3}z^{7}-18az^{7}-7z^{7}a^{-1}+4z^{7}a^{-3}+a^{6}z^{6}-4a^{4}z^{6}-4a^{2}z^{6}-8z^{6}a^{-2}+3z^{6}a^{-4}-10z^{6}-7a^{5}z^{5}+6a^{3}z^{5}+33az^{5}+11z^{5}a^{-1}-8z^{5}a^{-3}+z^{5}a^{-5}-4a^{6}z^{4}-2a^{4}z^{4}+8a^{2}z^{4}+5z^{4}a^{-2}-7z^{4}a^{-4}+18z^{4}+6a^{5}z^{3}-7a^{3}z^{3}-25az^{3}-7z^{3}a^{-1}+3z^{3}a^{-3}-2z^{3}a^{-5}+4a^{6}z^{2}+4a^{4}z^{2}-6a^{2}z^{2}+z^{2}a^{-2}+3z^{2}a^{-4}-8z^{2}-a^{5}z+3a^{3}z+6az+2za^{-1}-a^{6}-a^{4}-a^{-2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a119,}

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["K11a98"];

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]=

{ $4t^{2}-19t+31-19t^{-1}+4t^{-2}$, $-q^{5}+3q^{4}-5q^{3}+9q^{2}-11q+12-12q^{-1}+10q^{-2}-7q^{-3}+4q^{-4}-2q^{-5}+q^{-6}$ }

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]=

{K11a119,}

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:

(-3, 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 $-12$ $24$ $72$ $34$ $6$ $-288$ $-368$ $-160$ $24$ $-288$ $288$ $-408$ $-72$ ${\frac {4049}{10}}$ $-{\frac {1054}{15}}$ ${\frac {5018}{15}}$ $-{\frac {241}{6}}$ ${\frac {529}{10}}$

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 K11a98. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a97 K11a99