K11n145

K11n144

K11n146

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n145 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X14,4,15,3 X5,11,6,10 X7,20,8,21 X9,1,10,22 X18,11,19,12 X2,14,3,13 X15,9,16,8 X17,6,18,7 X12,19,13,20 X21,16,22,17
Gauss code	1, -7, 2, -1, -3, 9, -4, 8, -5, 3, 6, -10, 7, -2, -8, 11, -9, -6, 10, 4, -11, 5
Dowker-Thistlethwaite code	4 14 -10 -20 -22 18 2 -8 -6 12 -16

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:K11n145/ThurstonBennequinNumber Hyperbolic Volume 9.01207 A-Polynomial See Data:K11n145/A-polynomial

[edit Notes for K11n145's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[0,3]$ Rasmussen s-Invariant 0

[edit Notes for K11n145's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-t^{2}-3t+7-3t^{-1}-t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+5z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 9, 0 }
Jones polynomial	$-q^{5}+2q^{4}-2q^{3}+2q^{2}-2q+2+q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}-a^{2}z^{4}+6z^{4}-4a^{2}z^{2}-2z^{2}a^{-2}-z^{2}a^{-4}+9z^{2}-2a^{2}-2a^{-2}+5$
Kauffman polynomial (db, data sources)	$a^{2}z^{8}+z^{8}+a^{3}z^{7}+az^{7}+z^{7}a^{-1}+z^{7}a^{-3}-7a^{2}z^{6}+z^{6}a^{-2}+2z^{6}a^{-4}-8z^{6}-6a^{3}z^{5}-8az^{5}-6z^{5}a^{-1}-3z^{5}a^{-3}+z^{5}a^{-5}+14a^{2}z^{4}-2z^{4}a^{-2}-7z^{4}a^{-4}+19z^{4}+9a^{3}z^{3}+15az^{3}+10z^{3}a^{-1}+z^{3}a^{-3}-3z^{3}a^{-5}-10a^{2}z^{2}-2z^{2}a^{-2}+4z^{2}a^{-4}-16z^{2}-3a^{3}z-6az-4za^{-1}+za^{-5}+2a^{2}+2a^{-2}+5$
The A2 invariant	$-q^{12}-q^{10}+q^{6}+2q^{2}+2+q^{-2}+q^{-4}-q^{-6}-q^{-10}+q^{-14}-q^{-16}$
The G2 invariant	Data:K11n145/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n145 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["K11n145"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-t^{2}-3t+7-3t^{-1}-t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{6}+5z^{4}+2z^{2}+1$

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]=

{ 9, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}-a^{2}z^{4}+6z^{4}-4a^{2}z^{2}-2z^{2}a^{-2}-z^{2}a^{-4}+9z^{2}-2a^{2}-2a^{-2}+5$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{2}z^{8}+z^{8}+a^{3}z^{7}+az^{7}+z^{7}a^{-1}+z^{7}a^{-3}-7a^{2}z^{6}+z^{6}a^{-2}+2z^{6}a^{-4}-8z^{6}-6a^{3}z^{5}-8az^{5}-6z^{5}a^{-1}-3z^{5}a^{-3}+z^{5}a^{-5}+14a^{2}z^{4}-2z^{4}a^{-2}-7z^{4}a^{-4}+19z^{4}+9a^{3}z^{3}+15az^{3}+10z^{3}a^{-1}+z^{3}a^{-3}-3z^{3}a^{-5}-10a^{2}z^{2}-2z^{2}a^{-2}+4z^{2}a^{-4}-16z^{2}-3a^{3}z-6az-4za^{-1}+za^{-5}+2a^{2}+2a^{-2}+5$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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["K11n145"];

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]=

{ $t^{3}-t^{2}-3t+7-3t^{-1}-t^{-2}+t^{-3}$, $-q^{5}+2q^{4}-2q^{3}+2q^{2}-2q+2+q^{-3}-q^{-4}$ }

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]=

{}

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, 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 $8$ $0$ $32$ ${\frac {28}{3}}$ $-{\frac {76}{3}}$ $0$ $-32$ $0$ $-32$ ${\frac {256}{3}}$ $0$ ${\frac {224}{3}}$ $-{\frac {608}{3}}$ ${\frac {511}{15}}$ ${\frac {1132}{5}}$ $-{\frac {19316}{45}}$ ${\frac {401}{9}}$ $-{\frac {2129}{15}}$

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 K11n145. 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 χ 11                     1 -1 9                   1   1 7                 1 1   0 5             1 2 1     0 3             1 1       0 1         1 3 2         0 -1       1 1 2           2 -3       1 1             0 -5   1 1 1               1 -7                       0 -9 1                     -1

Integral Khovanov Homology (db, data source)

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

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.

K11n144 K11n146