K11n58

K11n57

K11n59

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n58 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X9,17,10,16 X11,18,12,19 X13,20,14,21 X6,15,7,16 X17,1,18,22 X19,12,20,13 X21,10,22,11
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, -5, 11, -6, 10, -7, -3, 8, 5, -9, 6, -10, 7, -11, 9
Dowker-Thistlethwaite code	4 8 14 2 -16 -18 -20 6 -22 -12 -10

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:K11n58/ThurstonBennequinNumber Hyperbolic Volume 9.97833 A-Polynomial See Data:K11n58/A-polynomial

[edit Notes for K11n58'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 K11n58's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{3}-4t^{2}+8t-9+8t^{-1}-4t^{-2}+t^{-3}$
Conway polynomial	$z^{6}+2z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 35, -2 }
Jones polynomial	$-q^{4}+2q^{3}-3q^{2}+5q-5+6q^{-1}-5q^{-2}+4q^{-3}-3q^{-4}+q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}-2a^{2}z^{4}-z^{4}a^{-2}+5z^{4}+a^{4}z^{2}-6a^{2}z^{2}-3z^{2}a^{-2}+9z^{2}+a^{4}-4a^{2}-2a^{-2}+6$
Kauffman polynomial (db, data sources)	$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+2a^{3}z^{7}-az^{7}-2z^{7}a^{-1}+z^{7}a^{-3}+a^{4}z^{6}-6a^{2}z^{6}-10z^{6}a^{-2}-17z^{6}-5a^{3}z^{5}-5az^{5}-5z^{5}a^{-1}-5z^{5}a^{-3}+8a^{2}z^{4}+15z^{4}a^{-2}+23z^{4}+3a^{5}z^{3}+6a^{3}z^{3}+6az^{3}+10z^{3}a^{-1}+7z^{3}a^{-3}+a^{6}z^{2}-10a^{2}z^{2}-8z^{2}a^{-2}-17z^{2}-a^{5}z-3a^{3}z-4az-4za^{-1}-2za^{-3}+a^{4}+4a^{2}+2a^{-2}+6$
The A2 invariant	$q^{16}-q^{12}-2q^{8}+q^{2}+3+q^{-2}+2q^{-4}-q^{-8}-q^{-12}$
The G2 invariant	Data:K11n58/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n58 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["K11n58"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{3}-4t^{2}+8t-9+8t^{-1}-4t^{-2}+t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 35, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{4}+2q^{3}-3q^{2}+5q-5+6q^{-1}-5q^{-2}+4q^{-3}-3q^{-4}+q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}-2a^{2}z^{4}-z^{4}a^{-2}+5z^{4}+a^{4}z^{2}-6a^{2}z^{2}-3z^{2}a^{-2}+9z^{2}+a^{4}-4a^{2}-2a^{-2}+6$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+2a^{3}z^{7}-az^{7}-2z^{7}a^{-1}+z^{7}a^{-3}+a^{4}z^{6}-6a^{2}z^{6}-10z^{6}a^{-2}-17z^{6}-5a^{3}z^{5}-5az^{5}-5z^{5}a^{-1}-5z^{5}a^{-3}+8a^{2}z^{4}+15z^{4}a^{-2}+23z^{4}+3a^{5}z^{3}+6a^{3}z^{3}+6az^{3}+10z^{3}a^{-1}+7z^{3}a^{-3}+a^{6}z^{2}-10a^{2}z^{2}-8z^{2}a^{-2}-17z^{2}-a^{5}z-3a^{3}z-4az-4za^{-1}-2za^{-3}+a^{4}+4a^{2}+2a^{-2}+6$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {8_16, 10_156, K11n15, K11n56,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n56,}

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["K11n58"];

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]=

{ $t^{3}-4t^{2}+8t-9+8t^{-1}-4t^{-2}+t^{-3}$, $-q^{4}+2q^{3}-3q^{2}+5q-5+6q^{-1}-5q^{-2}+4q^{-3}-3q^{-4}+q^{-5}$ }

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]=

{8_16, 10_156, K11n15, K11n56,}

In[6]:=

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

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

Out[6]=

{K11n56,}

Vassiliev invariants

V2 and V3:

(1, 1)

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 $4$ $8$ $8$ $-{\frac {34}{3}}$ $-{\frac {38}{3}}$ $32$ ${\frac {176}{3}}$ $-{\frac {64}{3}}$ $40$ ${\frac {32}{3}}$ $32$ $-{\frac {136}{3}}$ $-{\frac {152}{3}}$ $-{\frac {209}{30}}$ ${\frac {1258}{15}}$ $-{\frac {3898}{45}}$ $-{\frac {463}{18}}$ $-{\frac {1169}{30}}$

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 K11n58. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n57 K11n59