K11n172

K11n171

K11n173

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n172 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X10,3,11,4 X5,16,6,17 X12,8,13,7 X20,10,21,9 X2,11,3,12 X18,13,19,14 X15,4,16,5 X22,17,1,18 X8,20,9,19 X14,21,15,22
Gauss code	1, -6, 2, 8, -3, -1, 4, -10, 5, -2, 6, -4, 7, -11, -8, 3, 9, -7, 10, -5, 11, -9
Dowker-Thistlethwaite code	6 10 -16 12 20 2 18 -4 22 8 14

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n172/ThurstonBennequinNumber Hyperbolic Volume 13.3915 A-Polynomial See Data:K11n172/A-polynomial

[edit Notes for K11n172's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[0,3]$ Rasmussen s-Invariant 0

[edit Notes for K11n172's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+5t^{2}-11t+15-11t^{-1}+5t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-z^{4}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 49, 0 }
Jones polynomial	$q^{2}-3q+6-7q^{-1}+8q^{-2}-8q^{-3}+7q^{-4}-5q^{-5}+3q^{-6}-q^{-7}$
HOMFLY-PT polynomial (db, data sources)	$-z^{2}a^{6}-a^{6}+2z^{4}a^{4}+5z^{2}a^{4}+3a^{4}-z^{6}a^{2}-4z^{4}a^{2}-6z^{2}a^{2}-3a^{2}+z^{4}+2z^{2}+2$
Kauffman polynomial (db, data sources)	$2a^{5}z^{9}+2a^{3}z^{9}+3a^{6}z^{8}+8a^{4}z^{8}+5a^{2}z^{8}+a^{7}z^{7}-4a^{5}z^{7}-a^{3}z^{7}+4az^{7}-13a^{6}z^{6}-32a^{4}z^{6}-18a^{2}z^{6}+z^{6}-4a^{7}z^{5}-8a^{5}z^{5}-15a^{3}z^{5}-11az^{5}+16a^{6}z^{4}+36a^{4}z^{4}+22a^{2}z^{4}+2z^{4}+5a^{7}z^{3}+15a^{5}z^{3}+17a^{3}z^{3}+10az^{3}+3z^{3}a^{-1}-6a^{6}z^{2}-16a^{4}z^{2}-14a^{2}z^{2}+z^{2}a^{-2}-3z^{2}-2a^{7}z-5a^{5}z-5a^{3}z-3az-za^{-1}+a^{6}+3a^{4}+3a^{2}+2$
The A2 invariant	$-q^{22}+q^{18}-q^{16}+2q^{14}+q^{8}-2q^{6}+q^{4}-q^{2}+1+2q^{-2}-q^{-4}+q^{-6}$
The G2 invariant	Data:K11n172/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n172 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["K11n172"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+5t^{2}-11t+15-11t^{-1}+5t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 49, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

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

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2a^{5}z^{9}+2a^{3}z^{9}+3a^{6}z^{8}+8a^{4}z^{8}+5a^{2}z^{8}+a^{7}z^{7}-4a^{5}z^{7}-a^{3}z^{7}+4az^{7}-13a^{6}z^{6}-32a^{4}z^{6}-18a^{2}z^{6}+z^{6}-4a^{7}z^{5}-8a^{5}z^{5}-15a^{3}z^{5}-11az^{5}+16a^{6}z^{4}+36a^{4}z^{4}+22a^{2}z^{4}+2z^{4}+5a^{7}z^{3}+15a^{5}z^{3}+17a^{3}z^{3}+10az^{3}+3z^{3}a^{-1}-6a^{6}z^{2}-16a^{4}z^{2}-14a^{2}z^{2}+z^{2}a^{-2}-3z^{2}-2a^{7}z-5a^{5}z-5a^{3}z-3az-za^{-1}+a^{6}+3a^{4}+3a^{2}+2$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_27, K11n4, K11n21,}

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["K11n172"];

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]=

{ $-t^{3}+5t^{2}-11t+15-11t^{-1}+5t^{-2}-t^{-3}$, $q^{2}-3q+6-7q^{-1}+8q^{-2}-8q^{-3}+7q^{-4}-5q^{-5}+3q^{-6}-q^{-7}$ }

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]=

{9_27, K11n4, K11n21,}

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:

(0, -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 $0$ $-8$ $0$ $32$ $8$ $0$ $-{\frac {80}{3}}$ $-{\frac {32}{3}}$ $24$ $0$ $32$ $0$ $0$ $32$ ${\frac {760}{3}}$ $-{\frac {776}{3}}$ $-{\frac {128}{3}}$ $-80$

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=$0 is the signature of K11n172. 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 5                   1 1 3                 2   -2 1               4 1   3 -1             4 3     -1 -3           4 3       1 -5         4 4         0 -7       3 4           -1 -9     2 4             2 -11   1 3               -2 -13   2                 2 -15 1                   -1

Integral Khovanov Homology (db, data source)

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

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.

K11n171 K11n173