K11n171

K11n170

K11n172

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n171 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X3,11,4,10 X5,17,6,16 X12,8,13,7 X9,19,10,18 X11,3,12,2 X22,14,1,13 X20,16,21,15 X17,5,18,4 X19,9,20,8 X14,22,15,21
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 $\{2,3\}$ 3-genus 2 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n171/ThurstonBennequinNumber Hyperbolic Volume 13.6066 A-Polynomial See Data:K11n171/A-polynomial

[edit Notes for K11n171's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n171's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$6t^{2}-16t+21-16t^{-1}+6t^{-2}$
Conway polynomial	$6z^{4}+8z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 65, 4 }
Jones polynomial	$-2q^{11}+4q^{10}-7q^{9}+10q^{8}-11q^{7}+11q^{6}-9q^{5}+7q^{4}-3q^{3}+q^{2}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}a^{-4}+3z^{4}a^{-6}+2z^{4}a^{-8}+z^{2}a^{-4}+6z^{2}a^{-6}+3z^{2}a^{-8}-2z^{2}a^{-10}+2a^{-6}+a^{-8}-2a^{-10}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-9}+z^{9}a^{-11}+4z^{8}a^{-8}+5z^{8}a^{-10}+z^{8}a^{-12}+6z^{7}a^{-7}+6z^{7}a^{-9}+6z^{6}a^{-6}-3z^{6}a^{-8}-9z^{6}a^{-10}+3z^{5}a^{-5}-7z^{5}a^{-7}-12z^{5}a^{-9}+z^{5}a^{-11}+3z^{5}a^{-13}+z^{4}a^{-4}-9z^{4}a^{-6}+z^{4}a^{-8}+7z^{4}a^{-10}-4z^{4}a^{-12}-2z^{3}a^{-5}+2z^{3}a^{-7}+5z^{3}a^{-9}-7z^{3}a^{-11}-8z^{3}a^{-13}-z^{2}a^{-4}+7z^{2}a^{-6}-2z^{2}a^{-8}-7z^{2}a^{-10}+3z^{2}a^{-12}+za^{-7}-za^{-9}+3za^{-11}+5za^{-13}-2a^{-6}+a^{-8}+2a^{-10}$
The A2 invariant	Data:K11n171/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n171/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n171 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["K11n171"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$6t^{2}-16t+21-16t^{-1}+6t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$6z^{4}+8z^{2}+1$

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]=

{ 65, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{4}a^{-4}+3z^{4}a^{-6}+2z^{4}a^{-8}+z^{2}a^{-4}+6z^{2}a^{-6}+3z^{2}a^{-8}-2z^{2}a^{-10}+2a^{-6}+a^{-8}-2a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-9}+z^{9}a^{-11}+4z^{8}a^{-8}+5z^{8}a^{-10}+z^{8}a^{-12}+6z^{7}a^{-7}+6z^{7}a^{-9}+6z^{6}a^{-6}-3z^{6}a^{-8}-9z^{6}a^{-10}+3z^{5}a^{-5}-7z^{5}a^{-7}-12z^{5}a^{-9}+z^{5}a^{-11}+3z^{5}a^{-13}+z^{4}a^{-4}-9z^{4}a^{-6}+z^{4}a^{-8}+7z^{4}a^{-10}-4z^{4}a^{-12}-2z^{3}a^{-5}+2z^{3}a^{-7}+5z^{3}a^{-9}-7z^{3}a^{-11}-8z^{3}a^{-13}-z^{2}a^{-4}+7z^{2}a^{-6}-2z^{2}a^{-8}-7z^{2}a^{-10}+3z^{2}a^{-12}+za^{-7}-za^{-9}+3za^{-11}+5za^{-13}-2a^{-6}+a^{-8}+2a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a238,}

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["K11n171"];

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]=

{ $6t^{2}-16t+21-16t^{-1}+6t^{-2}$, $-2q^{11}+4q^{10}-7q^{9}+10q^{8}-11q^{7}+11q^{6}-9q^{5}+7q^{4}-3q^{3}+q^{2}$ }

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]=

{K11a238,}

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:

(8, 22)

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 $32$ $176$ $512$ ${\frac {3808}{3}}$ ${\frac {608}{3}}$ $5632$ ${\frac {30272}{3}}$ ${\frac {5312}{3}}$ $1424$ ${\frac {16384}{3}}$ $15488$ ${\frac {121856}{3}}$ ${\frac {19456}{3}}$ ${\frac {1221364}{15}}$ ${\frac {5568}{5}}$ ${\frac {1494256}{45}}$ ${\frac {5948}{9}}$ ${\frac {66724}{15}}$

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=$4 is the signature of K11n171. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ 0 1 2 3 4 5 6 7 8 9 χ 23                   2 -2 21                 2   2 19               5 2   -3 17             5 2     3 15           6 5       -1 13         5 5         0 11       4 6           2 9     3 5             -2 7     4               4 5 1 3                 -2 3 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n170 K11n172