K11a198

K11a197

K11a199

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a198 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a198/ThurstonBennequinNumber Hyperbolic Volume 15.066 A-Polynomial See Data:K11a198/A-polynomial

[edit Notes for K11a198's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for K11a198's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-11t^{2}+27t-35+27t^{-1}-11t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 115, 2 }
Jones polynomial	$q^{7}-4q^{6}+8q^{5}-13q^{4}+17q^{3}-18q^{2}+18q-15+11q^{-1}-6q^{-2}+3q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-2}+z^{6}-a^{2}z^{4}+z^{4}a^{-2}-2z^{4}a^{-4}+3z^{4}-2a^{2}z^{2}-z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}a^{-6}+5z^{2}-a^{2}-a^{-2}+3$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}+3az^{9}+8z^{9}a^{-1}+5z^{9}a^{-3}+3a^{2}z^{8}+16z^{8}a^{-2}+10z^{8}a^{-4}+9z^{8}+a^{3}z^{7}-6az^{7}-12z^{7}a^{-1}+6z^{7}a^{-3}+11z^{7}a^{-5}-12a^{2}z^{6}-47z^{6}a^{-2}-12z^{6}a^{-4}+8z^{6}a^{-6}-39z^{6}-4a^{3}z^{5}-5az^{5}-15z^{5}a^{-1}-32z^{5}a^{-3}-14z^{5}a^{-5}+4z^{5}a^{-7}+16a^{2}z^{4}+38z^{4}a^{-2}+z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+45z^{4}+5a^{3}z^{3}+14az^{3}+25z^{3}a^{-1}+24z^{3}a^{-3}+6z^{3}a^{-5}-2z^{3}a^{-7}-8a^{2}z^{2}-12z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}a^{-6}-19z^{2}-2a^{3}z-6az-8za^{-1}-5za^{-3}-za^{-5}+a^{2}+a^{-2}+3$
The A2 invariant	Data:K11a198/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11a198/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11a198 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["K11a198"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-11t^{2}+27t-35+27t^{-1}-11t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{6}+z^{4}+z^{2}+1$

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]=

{ 115, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-4q^{6}+8q^{5}-13q^{4}+17q^{3}-18q^{2}+18q-15+11q^{-1}-6q^{-2}+3q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-2}+z^{6}-a^{2}z^{4}+z^{4}a^{-2}-2z^{4}a^{-4}+3z^{4}-2a^{2}z^{2}-z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}a^{-6}+5z^{2}-a^{2}-a^{-2}+3$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}+3az^{9}+8z^{9}a^{-1}+5z^{9}a^{-3}+3a^{2}z^{8}+16z^{8}a^{-2}+10z^{8}a^{-4}+9z^{8}+a^{3}z^{7}-6az^{7}-12z^{7}a^{-1}+6z^{7}a^{-3}+11z^{7}a^{-5}-12a^{2}z^{6}-47z^{6}a^{-2}-12z^{6}a^{-4}+8z^{6}a^{-6}-39z^{6}-4a^{3}z^{5}-5az^{5}-15z^{5}a^{-1}-32z^{5}a^{-3}-14z^{5}a^{-5}+4z^{5}a^{-7}+16a^{2}z^{4}+38z^{4}a^{-2}+z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+45z^{4}+5a^{3}z^{3}+14az^{3}+25z^{3}a^{-1}+24z^{3}a^{-3}+6z^{3}a^{-5}-2z^{3}a^{-7}-8a^{2}z^{2}-12z^{2}a^{-2}+z^{2}a^{-4}+2z^{2}a^{-6}-19z^{2}-2a^{3}z-6az-8za^{-1}-5za^{-3}-za^{-5}+a^{2}+a^{-2}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_121, K11a41, K11a183, K11a331,}

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["K11a198"];

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]=

{ $2t^{3}-11t^{2}+27t-35+27t^{-1}-11t^{-2}+2t^{-3}$, $q^{7}-4q^{6}+8q^{5}-13q^{4}+17q^{3}-18q^{2}+18q-15+11q^{-1}-6q^{-2}+3q^{-3}-q^{-4}$ }

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]=

{10_121, K11a41, K11a183, K11a331,}

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:

(1, 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 $4$ $0$ $8$ ${\frac {14}{3}}$ $-{\frac {14}{3}}$ $0$ $32$ $32$ $0$ ${\frac {32}{3}}$ $0$ ${\frac {56}{3}}$ $-{\frac {56}{3}}$ ${\frac {1471}{30}}$ $-{\frac {302}{15}}$ ${\frac {2462}{45}}$ $-{\frac {223}{18}}$ ${\frac {991}{30}}$

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=$2 is the signature of K11a198. 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 15                       1 1 13                     3   -3 11                   5 1   4 9                 8 3     -5 7               9 5       4 5             9 8         -1 3           9 9           0 1         7 10             3 -1       4 8               -4 -3     2 7                 5 -5   1 4                   -3 -7   2                     2 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{9}$ $r=1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=3$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a197 K11a199