K11a47

K11a46

K11a48

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a47 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X14,5,15,6 X2837 X20,9,21,10 X22,11,1,12 X18,14,19,13 X6,15,7,16 X12,18,13,17 X16,20,17,19 X10,21,11,22
Gauss code	1, -4, 2, -1, 3, -8, 4, -2, 5, -11, 6, -9, 7, -3, 8, -10, 9, -7, 10, -5, 11, -6
Dowker-Thistlethwaite code	4 8 14 2 20 22 18 6 12 16 10

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 4 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a47/ThurstonBennequinNumber Hyperbolic Volume 15.5977 A-Polynomial See Data:K11a47/A-polynomial

[edit Notes for K11a47's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[2,4]$ Rasmussen s-Invariant 0

[edit Notes for K11a47's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-5t^{3}+14t^{2}-24t+29-24t^{-1}+14t^{-2}-5t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+3z^{6}+4z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{t^{2}-t+1\right\}$
Determinant and Signature	{ 117, 0 }
Jones polynomial	$q^{6}-3q^{5}+6q^{4}-12q^{3}+16q^{2}-18q+20-16q^{-1}+13q^{-2}-8q^{-3}+3q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-9z^{4}a^{-2}+z^{4}a^{-4}+16z^{4}-7a^{2}z^{2}-15z^{2}a^{-2}+3z^{2}a^{-4}+22z^{2}-5a^{2}-9a^{-2}+2a^{-4}+13$
Kauffman polynomial (db, data sources)	$z^{10}a^{-2}+z^{10}+4az^{9}+7z^{9}a^{-1}+3z^{9}a^{-3}+7a^{2}z^{8}+10z^{8}a^{-2}+4z^{8}a^{-4}+13z^{8}+6a^{3}z^{7}+5az^{7}-2z^{7}a^{-1}+2z^{7}a^{-3}+3z^{7}a^{-5}+3a^{4}z^{6}-11a^{2}z^{6}-28z^{6}a^{-2}-8z^{6}a^{-4}+z^{6}a^{-6}-33z^{6}+a^{5}z^{5}-10a^{3}z^{5}-25az^{5}-27z^{5}a^{-1}-22z^{5}a^{-3}-9z^{5}a^{-5}-4a^{4}z^{4}+11a^{2}z^{4}+28z^{4}a^{-2}+3z^{4}a^{-4}-3z^{4}a^{-6}+37z^{4}-2a^{5}z^{3}+8a^{3}z^{3}+31az^{3}+42z^{3}a^{-1}+30z^{3}a^{-3}+9z^{3}a^{-5}+a^{4}z^{2}-11a^{2}z^{2}-19z^{2}a^{-2}-z^{2}a^{-4}+2z^{2}a^{-6}-28z^{2}+a^{5}z-4a^{3}z-15az-21za^{-1}-15za^{-3}-4za^{-5}+5a^{2}+9a^{-2}+2a^{-4}+13$
The A2 invariant	$-q^{14}+q^{12}-4q^{10}-q^{4}+8q^{2}+1+6q^{-2}-q^{-4}-3q^{-6}-5q^{-10}+q^{-12}+q^{-18}$
The G2 invariant	$q^{80}-2q^{78}+5q^{76}-8q^{74}+10q^{72}-10q^{70}+4q^{68}+11q^{66}-32q^{64}+56q^{62}-76q^{60}+72q^{58}-44q^{56}-17q^{54}+110q^{52}-199q^{50}+259q^{48}-246q^{46}+131q^{44}+52q^{42}-273q^{40}+435q^{38}-479q^{36}+360q^{34}-112q^{32}-197q^{30}+437q^{28}-512q^{26}+392q^{24}-123q^{22}-181q^{20}+367q^{18}-368q^{16}+183q^{14}+119q^{12}-385q^{10}+511q^{8}-392q^{6}+97q^{4}+299q^{2}-617+749q^{-2}-608q^{-4}+270q^{-6}+167q^{-8}-537q^{-10}+735q^{-12}-661q^{-14}+376q^{-16}+9q^{-18}-349q^{-20}+498q^{-22}-427q^{-24}+162q^{-26}+140q^{-28}-358q^{-30}+388q^{-32}-232q^{-34}-64q^{-36}+347q^{-38}-505q^{-40}+463q^{-42}-264q^{-44}-42q^{-46}+310q^{-48}-453q^{-50}+451q^{-52}-307q^{-54}+106q^{-56}+93q^{-58}-223q^{-60}+258q^{-62}-217q^{-64}+131q^{-66}-34q^{-68}-36q^{-70}+74q^{-72}-78q^{-74}+63q^{-76}-35q^{-78}+13q^{-80}+3q^{-82}-12q^{-84}+10q^{-86}-9q^{-88}+5q^{-90}-2q^{-92}+q^{-94}$

Further Quantum Invariants

K11a47 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["K11a47"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-5t^{3}+14t^{2}-24t+29-24t^{-1}+14t^{-2}-5t^{-3}+t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\left\{t^{2}-t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 117, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-3q^{5}+6q^{4}-12q^{3}+16q^{2}-18q+20-16q^{-1}+13q^{-2}-8q^{-3}+3q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+6z^{6}-4a^{2}z^{4}-9z^{4}a^{-2}+z^{4}a^{-4}+16z^{4}-7a^{2}z^{2}-15z^{2}a^{-2}+3z^{2}a^{-4}+22z^{2}-5a^{2}-9a^{-2}+2a^{-4}+13$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$z^{10}a^{-2}+z^{10}+4az^{9}+7z^{9}a^{-1}+3z^{9}a^{-3}+7a^{2}z^{8}+10z^{8}a^{-2}+4z^{8}a^{-4}+13z^{8}+6a^{3}z^{7}+5az^{7}-2z^{7}a^{-1}+2z^{7}a^{-3}+3z^{7}a^{-5}+3a^{4}z^{6}-11a^{2}z^{6}-28z^{6}a^{-2}-8z^{6}a^{-4}+z^{6}a^{-6}-33z^{6}+a^{5}z^{5}-10a^{3}z^{5}-25az^{5}-27z^{5}a^{-1}-22z^{5}a^{-3}-9z^{5}a^{-5}-4a^{4}z^{4}+11a^{2}z^{4}+28z^{4}a^{-2}+3z^{4}a^{-4}-3z^{4}a^{-6}+37z^{4}-2a^{5}z^{3}+8a^{3}z^{3}+31az^{3}+42z^{3}a^{-1}+30z^{3}a^{-3}+9z^{3}a^{-5}+a^{4}z^{2}-11a^{2}z^{2}-19z^{2}a^{-2}-z^{2}a^{-4}+2z^{2}a^{-6}-28z^{2}+a^{5}z-4a^{3}z-15az-21za^{-1}-15za^{-3}-4za^{-5}+5a^{2}+9a^{-2}+2a^{-4}+13$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a44, K11a109,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a44,}

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["K11a47"];

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]=

{ $t^{4}-5t^{3}+14t^{2}-24t+29-24t^{-1}+14t^{-2}-5t^{-3}+t^{-4}$, $q^{6}-3q^{5}+6q^{4}-12q^{3}+16q^{2}-18q+20-16q^{-1}+13q^{-2}-8q^{-3}+3q^{-4}-q^{-5}$ }

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]=

{K11a44, K11a109,}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11a44,}

Vassiliev invariants

V2 and V3:

(3, -2)

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 $12$ $-16$ $72$ $62$ $2$ $-192$ $-{\frac {832}{3}}$ $-{\frac {64}{3}}$ $-48$ $288$ $128$ $744$ $24$ ${\frac {8751}{10}}$ $-{\frac {106}{15}}$ ${\frac {1514}{5}}$ ${\frac {59}{2}}$ ${\frac {271}{10}}$

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 K11a47. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     2   -2 9                   4 1   3 7                 8 2     -6 5               8 4       4 3             10 8         -2 1           10 8           2 -1         7 11             4 -3       6 9               -3 -5     2 7                 5 -7   1 6                   -5 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{11}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a46 K11a48