K11n170

K11n169

K11n171

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n170 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,3,11,4 X16,5,17,6 X12,8,13,7 X18,9,19,10 X2,11,3,12 X13,22,14,1 X15,20,16,21 X4,17,5,18 X8,19,9,20 X21,14,22,15
Gauss code	1, -6, 2, -9, 3, -1, 4, -10, 5, -2, 6, -4, -7, 11, -8, -3, 9, -5, 10, 8, -11, 7
Dowker-Thistlethwaite code	6 10 16 12 18 2 -22 -20 4 8 -14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n170/ThurstonBennequinNumber Hyperbolic Volume 13.6098 A-Polynomial See Data:K11n170/A-polynomial

[edit Notes for K11n170's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n170's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+16t-25+16t^{-1}-3t^{-2}$
Conway polynomial	$-3z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 63, -2 }
Jones polynomial	$q-3+6q^{-1}-9q^{-2}+11q^{-3}-10q^{-4}+10q^{-5}-7q^{-6}+4q^{-7}-2q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-2a^{8}+4z^{2}a^{6}+3a^{6}-2z^{4}a^{4}-2z^{2}a^{4}-a^{4}-z^{4}a^{2}+z^{2}a^{2}+a^{2}+z^{2}$
Kauffman polynomial (db, data sources)	$3z^{5}a^{9}-8z^{3}a^{9}+5za^{9}+z^{8}a^{8}-4z^{4}a^{8}+3z^{2}a^{8}-2a^{8}+z^{9}a^{7}+z^{5}a^{7}-9z^{3}a^{7}+5za^{7}+5z^{8}a^{6}-9z^{6}a^{6}+3z^{4}a^{6}+2z^{2}a^{6}-3a^{6}+z^{9}a^{5}+6z^{7}a^{5}-15z^{5}a^{5}+10z^{3}a^{5}-za^{5}+4z^{8}a^{4}-4z^{6}a^{4}+3z^{2}a^{4}-a^{4}+6z^{7}a^{3}-10z^{5}a^{3}+8z^{3}a^{3}-za^{3}+5z^{6}a^{2}-6z^{4}a^{2}+3z^{2}a^{2}-a^{2}+3z^{5}a-3z^{3}a+z^{4}-z^{2}$
The A2 invariant	Data:K11n170/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n170/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n170 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["K11n170"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{2}+16t-25+16t^{-1}-3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 63, -2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q-3+6q^{-1}-9q^{-2}+11q^{-3}-10q^{-4}+10q^{-5}-7q^{-6}+4q^{-7}-2q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-2a^{8}+4z^{2}a^{6}+3a^{6}-2z^{4}a^{4}-2z^{2}a^{4}-a^{4}-z^{4}a^{2}+z^{2}a^{2}+a^{2}+z^{2}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

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

"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["K11n170"];

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]=

{ $-3t^{2}+16t-25+16t^{-1}-3t^{-2}$, $q-3+6q^{-1}-9q^{-2}+11q^{-3}-10q^{-4}+10q^{-5}-7q^{-6}+4q^{-7}-2q^{-8}$ }

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:

(4, -9)

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 $16$ $-72$ $128$ ${\frac {1208}{3}}$ ${\frac {256}{3}}$ $-1152$ $-2416$ $-384$ $-520$ ${\frac {2048}{3}}$ $2592$ ${\frac {19328}{3}}$ ${\frac {4096}{3}}$ ${\frac {216302}{15}}$ $-{\frac {20608}{15}}$ ${\frac {361088}{45}}$ ${\frac {2194}{9}}$ ${\frac {18782}{15}}$

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 $s=$-2 is the signature of K11n170. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 χ 3                   1 1 1                 2   -2 -1               4 1   3 -3             6 3     -3 -5           5 3       2 -7         5 6         1 -9       5 5           0 -11     2 5             3 -13   2 5               -3 -15   2                 2 -17 2                   -2

Integral Khovanov Homology (db, data source)

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

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.

K11n169 K11n171