K11a177

K11a176

K11a178

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a177 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a177/ThurstonBennequinNumber Hyperbolic Volume 13.045 A-Polynomial See Data:K11a177/A-polynomial

[edit Notes for K11a177's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a177's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+13t^{2}-19t+21-19t^{-1}+13t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+3z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 97, 4 }
Jones polynomial	$q^{10}-4q^{9}+7q^{8}-11q^{7}+14q^{6}-15q^{5}+15q^{4}-12q^{3}+9q^{2}-5q+3-q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-4}-z^{6}a^{-2}+6z^{6}a^{-4}-2z^{6}a^{-6}-4z^{4}a^{-2}+14z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}-4z^{2}a^{-2}+15z^{2}a^{-4}-9z^{2}a^{-6}+2z^{2}a^{-8}-a^{-2}+5a^{-4}-3a^{-6}$
Kauffman polynomial (db, data sources)	$z^{10}a^{-4}+z^{10}a^{-6}+3z^{9}a^{-3}+7z^{9}a^{-5}+4z^{9}a^{-7}+3z^{8}a^{-2}+6z^{8}a^{-4}+10z^{8}a^{-6}+7z^{8}a^{-8}+z^{7}a^{-1}-9z^{7}a^{-3}-18z^{7}a^{-5}+8z^{7}a^{-9}-13z^{6}a^{-2}-34z^{6}a^{-4}-36z^{6}a^{-6}-8z^{6}a^{-8}+7z^{6}a^{-10}-4z^{5}a^{-1}+3z^{5}a^{-3}+7z^{5}a^{-5}-12z^{5}a^{-7}-8z^{5}a^{-9}+4z^{5}a^{-11}+17z^{4}a^{-2}+47z^{4}a^{-4}+39z^{4}a^{-6}+z^{4}a^{-8}-7z^{4}a^{-10}+z^{4}a^{-12}+4z^{3}a^{-1}+6z^{3}a^{-3}+8z^{3}a^{-5}+8z^{3}a^{-7}-z^{3}a^{-9}-3z^{3}a^{-11}-8z^{2}a^{-2}-25z^{2}a^{-4}-18z^{2}a^{-6}+z^{2}a^{-10}-za^{-1}-3za^{-3}-5za^{-5}-za^{-7}+2za^{-9}+a^{-2}+5a^{-4}+3a^{-6}$
The A2 invariant	$-q^{2}+1-q^{-2}+q^{-4}+2q^{-6}-q^{-8}+4q^{-10}-2q^{-12}+2q^{-14}+q^{-16}-q^{-18}+2q^{-20}-3q^{-22}-q^{-26}-q^{-28}+q^{-30}$
The G2 invariant	$q^{12}-2q^{10}+5q^{8}-9q^{6}+10q^{4}-11q^{2}+3+14q^{-2}-35q^{-4}+56q^{-6}-66q^{-8}+50q^{-10}-12q^{-12}-51q^{-14}+116q^{-16}-156q^{-18}+154q^{-20}-93q^{-22}-8q^{-24}+123q^{-26}-205q^{-28}+225q^{-30}-168q^{-32}+57q^{-34}+71q^{-36}-167q^{-38}+196q^{-40}-142q^{-42}+46q^{-44}+69q^{-46}-136q^{-48}+131q^{-50}-62q^{-52}-50q^{-54}+153q^{-56}-199q^{-58}+170q^{-60}-60q^{-62}-87q^{-64}+224q^{-66}-291q^{-68}+264q^{-70}-149q^{-72}-16q^{-74}+171q^{-76}-265q^{-78}+267q^{-80}-178q^{-82}+44q^{-84}+84q^{-86}-158q^{-88}+150q^{-90}-84q^{-92}-12q^{-94}+86q^{-96}-112q^{-98}+76q^{-100}-q^{-102}-88q^{-104}+148q^{-106}-157q^{-108}+113q^{-110}-33q^{-112}-62q^{-114}+131q^{-116}-166q^{-118}+158q^{-120}-107q^{-122}+37q^{-124}+37q^{-126}-91q^{-128}+114q^{-130}-107q^{-132}+78q^{-134}-35q^{-136}-6q^{-138}+34q^{-140}-50q^{-142}+47q^{-144}-32q^{-146}+18q^{-148}-2q^{-150}-6q^{-152}+9q^{-154}-10q^{-156}+6q^{-158}-3q^{-160}+q^{-162}$

Further Quantum Invariants

K11a177 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["K11a177"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+13t^{2}-19t+21-19t^{-1}+13t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$z^{8}+3z^{6}+3z^{4}+4z^{2}+1$

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]=

{ 97, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{10}-4q^{9}+7q^{8}-11q^{7}+14q^{6}-15q^{5}+15q^{4}-12q^{3}+9q^{2}-5q+3-q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}a^{-4}-z^{6}a^{-2}+6z^{6}a^{-4}-2z^{6}a^{-6}-4z^{4}a^{-2}+14z^{4}a^{-4}-8z^{4}a^{-6}+z^{4}a^{-8}-4z^{2}a^{-2}+15z^{2}a^{-4}-9z^{2}a^{-6}+2z^{2}a^{-8}-a^{-2}+5a^{-4}-3a^{-6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-4}+z^{10}a^{-6}+3z^{9}a^{-3}+7z^{9}a^{-5}+4z^{9}a^{-7}+3z^{8}a^{-2}+6z^{8}a^{-4}+10z^{8}a^{-6}+7z^{8}a^{-8}+z^{7}a^{-1}-9z^{7}a^{-3}-18z^{7}a^{-5}+8z^{7}a^{-9}-13z^{6}a^{-2}-34z^{6}a^{-4}-36z^{6}a^{-6}-8z^{6}a^{-8}+7z^{6}a^{-10}-4z^{5}a^{-1}+3z^{5}a^{-3}+7z^{5}a^{-5}-12z^{5}a^{-7}-8z^{5}a^{-9}+4z^{5}a^{-11}+17z^{4}a^{-2}+47z^{4}a^{-4}+39z^{4}a^{-6}+z^{4}a^{-8}-7z^{4}a^{-10}+z^{4}a^{-12}+4z^{3}a^{-1}+6z^{3}a^{-3}+8z^{3}a^{-5}+8z^{3}a^{-7}-z^{3}a^{-9}-3z^{3}a^{-11}-8z^{2}a^{-2}-25z^{2}a^{-4}-18z^{2}a^{-6}+z^{2}a^{-10}-za^{-1}-3za^{-3}-5za^{-5}-za^{-7}+2za^{-9}+a^{-2}+5a^{-4}+3a^{-6}$

"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["K11a177"];

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]=

{ $t^{4}-5t^{3}+13t^{2}-19t+21-19t^{-1}+13t^{-2}-5t^{-3}+t^{-4}$, $q^{10}-4q^{9}+7q^{8}-11q^{7}+14q^{6}-15q^{5}+15q^{4}-12q^{3}+9q^{2}-5q+3-q^{-1}$ }

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, 7)

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 $16$ $56$ $128$ ${\frac {824}{3}}$ ${\frac {112}{3}}$ $896$ ${\frac {4496}{3}}$ ${\frac {896}{3}}$ $152$ ${\frac {2048}{3}}$ $1568$ ${\frac {13184}{3}}$ ${\frac {1792}{3}}$ ${\frac {124142}{15}}$ ${\frac {8032}{15}}$ ${\frac {129968}{45}}$ ${\frac {34}{9}}$ ${\frac {5582}{15}}$

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 K11a177. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11a176 K11a178