K11n178

K11n177

K11n179

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11n178 at Knotilus!

Knot presentations

Planar diagram presentation	X6271 X3,11,4,10 X16,6,17,5 X18,7,19,8 X20,10,21,9 X11,5,12,4 X8,14,9,13 X2,16,3,15 X22,17,1,18 X14,19,15,20 X12,22,13,21
Gauss code	1, -8, -2, 6, 3, -1, 4, -7, 5, 2, -6, -11, 7, -10, 8, -3, 9, -4, 10, -5, 11, -9
Dowker-Thistlethwaite code	6 -10 16 18 20 -4 8 2 22 14 12

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11n178/ThurstonBennequinNumber Hyperbolic Volume 16.9983 A-Polynomial See Data:K11n178/A-polynomial

[edit Notes for K11n178's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $3$ Rasmussen s-Invariant -2

[edit Notes for K11n178's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$2t^{3}-9t^{2}+22t-29+22t^{-1}-9t^{-2}+2t^{-3}$
Conway polynomial	$2z^{6}+3z^{4}+4z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 95, 2 }
Jones polynomial	$q^{9}-5q^{8}+9q^{7}-13q^{6}+16q^{5}-16q^{4}+15q^{3}-11q^{2}+7q-2$
HOMFLY-PT polynomial (db, data sources)	$2z^{6}a^{-4}-2z^{4}a^{-2}+8z^{4}a^{-4}-3z^{4}a^{-6}-3z^{2}a^{-2}+12z^{2}a^{-4}-6z^{2}a^{-6}+z^{2}a^{-8}-a^{-2}+5a^{-4}-3a^{-6}$
Kauffman polynomial (db, data sources)	$4z^{9}a^{-5}+4z^{9}a^{-7}+9z^{8}a^{-4}+17z^{8}a^{-6}+8z^{8}a^{-8}+6z^{7}a^{-3}+3z^{7}a^{-5}+2z^{7}a^{-7}+5z^{7}a^{-9}+z^{6}a^{-2}-22z^{6}a^{-4}-44z^{6}a^{-6}-20z^{6}a^{-8}+z^{6}a^{-10}-7z^{5}a^{-3}-20z^{5}a^{-5}-24z^{5}a^{-7}-11z^{5}a^{-9}+8z^{4}a^{-2}+31z^{4}a^{-4}+35z^{4}a^{-6}+11z^{4}a^{-8}-z^{4}a^{-10}+3z^{3}a^{-1}+9z^{3}a^{-3}+16z^{3}a^{-5}+14z^{3}a^{-7}+4z^{3}a^{-9}-6z^{2}a^{-2}-20z^{2}a^{-4}-15z^{2}a^{-6}-z^{2}a^{-8}-za^{-1}-3za^{-3}-5za^{-5}-za^{-7}+2za^{-9}+a^{-2}+5a^{-4}+3a^{-6}$
The A2 invariant	$-2+3q^{-2}-2q^{-4}+q^{-6}+4q^{-8}-2q^{-10}+4q^{-12}-2q^{-14}+2q^{-16}+q^{-18}-3q^{-20}+2q^{-22}-3q^{-24}-q^{-26}+q^{-28}$
The G2 invariant	Data:K11n178/QuantumInvariant/G2/1,0

Further Quantum Invariants

K11n178 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["K11n178"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$2t^{3}-9t^{2}+22t-29+22t^{-1}-9t^{-2}+2t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 95, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{9}-5q^{8}+9q^{7}-13q^{6}+16q^{5}-16q^{4}+15q^{3}-11q^{2}+7q-2$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$2z^{6}a^{-4}-2z^{4}a^{-2}+8z^{4}a^{-4}-3z^{4}a^{-6}-3z^{2}a^{-2}+12z^{2}a^{-4}-6z^{2}a^{-6}+z^{2}a^{-8}-a^{-2}+5a^{-4}-3a^{-6}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$4z^{9}a^{-5}+4z^{9}a^{-7}+9z^{8}a^{-4}+17z^{8}a^{-6}+8z^{8}a^{-8}+6z^{7}a^{-3}+3z^{7}a^{-5}+2z^{7}a^{-7}+5z^{7}a^{-9}+z^{6}a^{-2}-22z^{6}a^{-4}-44z^{6}a^{-6}-20z^{6}a^{-8}+z^{6}a^{-10}-7z^{5}a^{-3}-20z^{5}a^{-5}-24z^{5}a^{-7}-11z^{5}a^{-9}+8z^{4}a^{-2}+31z^{4}a^{-4}+35z^{4}a^{-6}+11z^{4}a^{-8}-z^{4}a^{-10}+3z^{3}a^{-1}+9z^{3}a^{-3}+16z^{3}a^{-5}+14z^{3}a^{-7}+4z^{3}a^{-9}-6z^{2}a^{-2}-20z^{2}a^{-4}-15z^{2}a^{-6}-z^{2}a^{-8}-za^{-1}-3za^{-3}-5za^{-5}-za^{-7}+2za^{-9}+a^{-2}+5a^{-4}+3a^{-6}$

"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["K11n178"];

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]=

{ $2t^{3}-9t^{2}+22t-29+22t^{-1}-9t^{-2}+2t^{-3}$, $q^{9}-5q^{8}+9q^{7}-13q^{6}+16q^{5}-16q^{4}+15q^{3}-11q^{2}+7q-2$ }

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]=

{}

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 $16$ $56$ $128$ ${\frac {824}{3}}$ ${\frac {112}{3}}$ $896$ ${\frac {4496}{3}}$ ${\frac {800}{3}}$ $184$ ${\frac {2048}{3}}$ $1568$ ${\frac {13184}{3}}$ ${\frac {1792}{3}}$ ${\frac {124142}{15}}$ ${\frac {4192}{15}}$ ${\frac {138608}{45}}$ ${\frac {610}{9}}$ ${\frac {6062}{15}}$

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 K11n178. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

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

Integral Khovanov Homology (db, data source)

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

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.

K11n177 K11n179