K11n180

K11n179

K11n181

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n180 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X3,13,4,12 X5,17,6,16 X14,8,15,7 X9,19,10,18 X11,21,12,20 X13,3,14,2 X22,16,1,15 X17,5,18,4 X19,11,20,10 X21,9,22,8
Gauss code	1, 7, -2, 9, -3, -1, 4, 11, -5, 10, -6, 2, -7, -4, 8, 3, -9, 5, -10, 6, -11, -8
Dowker-Thistlethwaite code	6 -12 -16 14 -18 -20 -2 22 -4 -10 -8

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{3,4\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n180/ThurstonBennequinNumber Hyperbolic Volume 12.6965 A-Polynomial See Data:K11n180/A-polynomial

[edit Notes for K11n180's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -6

[edit Notes for K11n180's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$3t^{3}-7t^{2}+11t-13+11t^{-1}-7t^{-2}+3t^{-3}$
Conway polynomial	$3z^{6}+11z^{4}+10z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 55, 6 }
Jones polynomial	$-2q^{12}+4q^{11}-7q^{10}+9q^{9}-9q^{8}+9q^{7}-7q^{6}+5q^{5}-2q^{4}+q^{3}$
HOMFLY-PT polynomial (db, data sources)	$z^{6}a^{-6}+2z^{6}a^{-8}+4z^{4}a^{-6}+9z^{4}a^{-8}-2z^{4}a^{-10}+4z^{2}a^{-6}+12z^{2}a^{-8}-6z^{2}a^{-10}+5a^{-8}-4a^{-10}$
Kauffman polynomial (db, data sources)	$z^{9}a^{-9}+z^{9}a^{-11}+3z^{8}a^{-8}+5z^{8}a^{-10}+2z^{8}a^{-12}+2z^{7}a^{-7}+2z^{7}a^{-9}+2z^{7}a^{-11}+2z^{7}a^{-13}+z^{6}a^{-6}-11z^{6}a^{-8}-13z^{6}a^{-10}+z^{6}a^{-14}-6z^{5}a^{-7}-10z^{5}a^{-9}-4z^{5}a^{-11}-4z^{4}a^{-6}+17z^{4}a^{-8}+16z^{4}a^{-10}-3z^{4}a^{-12}+2z^{4}a^{-14}+4z^{3}a^{-7}+7z^{3}a^{-9}+3z^{3}a^{-15}+4z^{2}a^{-6}-16z^{2}a^{-8}-15z^{2}a^{-10}+4z^{2}a^{-12}-z^{2}a^{-14}-3za^{-9}+za^{-11}+za^{-13}-3za^{-15}+5a^{-8}+4a^{-10}$
The A2 invariant	Data:K11n180/QuantumInvariant/A2/1,0
The G2 invariant	Data:K11n180/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n180 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["K11n180"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$3t^{3}-7t^{2}+11t-13+11t^{-1}-7t^{-2}+3t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$3z^{6}+11z^{4}+10z^{2}+1$

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]=

{ 55, 6 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{6}a^{-6}+2z^{6}a^{-8}+4z^{4}a^{-6}+9z^{4}a^{-8}-2z^{4}a^{-10}+4z^{2}a^{-6}+12z^{2}a^{-8}-6z^{2}a^{-10}+5a^{-8}-4a^{-10}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{9}a^{-9}+z^{9}a^{-11}+3z^{8}a^{-8}+5z^{8}a^{-10}+2z^{8}a^{-12}+2z^{7}a^{-7}+2z^{7}a^{-9}+2z^{7}a^{-11}+2z^{7}a^{-13}+z^{6}a^{-6}-11z^{6}a^{-8}-13z^{6}a^{-10}+z^{6}a^{-14}-6z^{5}a^{-7}-10z^{5}a^{-9}-4z^{5}a^{-11}-4z^{4}a^{-6}+17z^{4}a^{-8}+16z^{4}a^{-10}-3z^{4}a^{-12}+2z^{4}a^{-14}+4z^{3}a^{-7}+7z^{3}a^{-9}+3z^{3}a^{-15}+4z^{2}a^{-6}-16z^{2}a^{-8}-15z^{2}a^{-10}+4z^{2}a^{-12}-z^{2}a^{-14}-3za^{-9}+za^{-11}+za^{-13}-3za^{-15}+5a^{-8}+4a^{-10}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a339,}

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["K11n180"];

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]=

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

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]=

{K11a339,}

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:

(10, 30)

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 $40$ $240$ $800$ ${\frac {5756}{3}}$ ${\frac {916}{3}}$ $9600$ $16864$ $3008$ $2288$ ${\frac {32000}{3}}$ $28800$ ${\frac {230240}{3}}$ ${\frac {36640}{3}}$ ${\frac {452311}{3}}$ $3516$ ${\frac {539404}{9}}$ ${\frac {8269}{9}}$ ${\frac {23863}{3}}$

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=$6 is the signature of K11n180. 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 25                   2 -2 23                 2   2 21               5 2   -3 19             4 2     2 17           5 5       0 15         4 4         0 13       3 5           2 11     2 4             -2 9     3               3 7 1 2                 -1 5 1                   1

Integral Khovanov Homology (db, data source)

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

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.

K11n179 K11n181