K11a147

K11a146

K11a148

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a147 at Knotilus!

Knot presentations

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

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a147/ThurstonBennequinNumber Hyperbolic Volume 17.2353 A-Polynomial See Data:K11a147/A-polynomial

[edit Notes for K11a147's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a147's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-17t^{2}+32t-39+32t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}-z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 151, 2 }
Jones polynomial	$-q^{8}+4q^{7}-10q^{6}+16q^{5}-21q^{4}+25q^{3}-24q^{2}+21q-15+9q^{-1}-4q^{-2}+q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-10z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-8z^{2}a^{-2}+9z^{2}a^{-4}-2z^{2}a^{-6}+3z^{2}-2a^{-2}+4a^{-4}-2a^{-6}+1$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}a^{-4}+6z^{9}a^{-1}+13z^{9}a^{-3}+7z^{9}a^{-5}+15z^{8}a^{-2}+19z^{8}a^{-4}+11z^{8}a^{-6}+7z^{8}+4az^{7}-4z^{7}a^{-1}-13z^{7}a^{-3}+4z^{7}a^{-5}+9z^{7}a^{-7}+a^{2}z^{6}-41z^{6}a^{-2}-45z^{6}a^{-4}-16z^{6}a^{-6}+4z^{6}a^{-8}-15z^{6}-9az^{5}-12z^{5}a^{-1}-15z^{5}a^{-3}-27z^{5}a^{-5}-14z^{5}a^{-7}+z^{5}a^{-9}-2a^{2}z^{4}+32z^{4}a^{-2}+34z^{4}a^{-4}+10z^{4}a^{-6}-4z^{4}a^{-8}+10z^{4}+6az^{3}+12z^{3}a^{-1}+19z^{3}a^{-3}+24z^{3}a^{-5}+10z^{3}a^{-7}-z^{3}a^{-9}+a^{2}z^{2}-12z^{2}a^{-2}-14z^{2}a^{-4}-5z^{2}a^{-6}+z^{2}a^{-8}-3z^{2}-az-3za^{-1}-5za^{-3}-7za^{-5}-4za^{-7}+2a^{-2}+4a^{-4}+2a^{-6}+1$
The A2 invariant	$q^{8}-2q^{6}+3q^{4}-2q^{2}-1+4q^{-2}-5q^{-4}+5q^{-6}-2q^{-8}+2q^{-10}+3q^{-12}-3q^{-14}+4q^{-16}-3q^{-18}-q^{-20}+q^{-22}-q^{-24}$
The G2 invariant	$q^{46}-3q^{44}+8q^{42}-16q^{40}+22q^{38}-25q^{36}+14q^{34}+18q^{32}-66q^{30}+128q^{28}-176q^{26}+175q^{24}-101q^{22}-63q^{20}+289q^{18}-494q^{16}+595q^{14}-503q^{12}+187q^{10}+276q^{8}-746q^{6}+1033q^{4}-982q^{2}+583+52q^{-2}-689q^{-4}+1072q^{-6}-1041q^{-8}+605q^{-10}+49q^{-12}-639q^{-14}+889q^{-16}-697q^{-18}+143q^{-20}+532q^{-22}-1009q^{-24}+1072q^{-26}-654q^{-28}-117q^{-30}+932q^{-32}-1485q^{-34}+1542q^{-36}-1054q^{-38}+200q^{-40}+732q^{-42}-1392q^{-44}+1563q^{-46}-1188q^{-48}+435q^{-50}+387q^{-52}-955q^{-54}+1059q^{-56}-686q^{-58}+53q^{-60}+563q^{-62}-860q^{-64}+723q^{-66}-231q^{-68}-416q^{-70}+919q^{-72}-1075q^{-74}+828q^{-76}-285q^{-78}-335q^{-80}+801q^{-82}-973q^{-84}+831q^{-86}-473q^{-88}+42q^{-90}+307q^{-92}-501q^{-94}+504q^{-96}-370q^{-98}+187q^{-100}-9q^{-102}-105q^{-104}+149q^{-106}-143q^{-108}+99q^{-110}-49q^{-112}+12q^{-114}+12q^{-116}-19q^{-118}+17q^{-120}-13q^{-122}+7q^{-124}-3q^{-126}+q^{-128}$

Further Quantum Invariants

K11a147 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["K11a147"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-17t^{2}+32t-39+32t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

{ 151, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+4q^{7}-10q^{6}+16q^{5}-21q^{4}+25q^{3}-24q^{2}+21q-15+9q^{-1}-4q^{-2}+q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-5z^{6}a^{-2}+2z^{6}a^{-4}+z^{6}-10z^{4}a^{-2}+7z^{4}a^{-4}-z^{4}a^{-6}+3z^{4}-8z^{2}a^{-2}+9z^{2}a^{-4}-2z^{2}a^{-6}+3z^{2}-2a^{-2}+4a^{-4}-2a^{-6}+1$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}a^{-4}+6z^{9}a^{-1}+13z^{9}a^{-3}+7z^{9}a^{-5}+15z^{8}a^{-2}+19z^{8}a^{-4}+11z^{8}a^{-6}+7z^{8}+4az^{7}-4z^{7}a^{-1}-13z^{7}a^{-3}+4z^{7}a^{-5}+9z^{7}a^{-7}+a^{2}z^{6}-41z^{6}a^{-2}-45z^{6}a^{-4}-16z^{6}a^{-6}+4z^{6}a^{-8}-15z^{6}-9az^{5}-12z^{5}a^{-1}-15z^{5}a^{-3}-27z^{5}a^{-5}-14z^{5}a^{-7}+z^{5}a^{-9}-2a^{2}z^{4}+32z^{4}a^{-2}+34z^{4}a^{-4}+10z^{4}a^{-6}-4z^{4}a^{-8}+10z^{4}+6az^{3}+12z^{3}a^{-1}+19z^{3}a^{-3}+24z^{3}a^{-5}+10z^{3}a^{-7}-z^{3}a^{-9}+a^{2}z^{2}-12z^{2}a^{-2}-14z^{2}a^{-4}-5z^{2}a^{-6}+z^{2}a^{-8}-3z^{2}-az-3za^{-1}-5za^{-3}-7za^{-5}-4za^{-7}+2a^{-2}+4a^{-4}+2a^{-6}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11a322,}

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["K11a147"];

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]=

{ $-t^{4}+6t^{3}-17t^{2}+32t-39+32t^{-1}-17t^{-2}+6t^{-3}-t^{-4}$, $-q^{8}+4q^{7}-10q^{6}+16q^{5}-21q^{4}+25q^{3}-24q^{2}+21q-15+9q^{-1}-4q^{-2}+q^{-3}$ }

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]=

{}

In[6]:=

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

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

Out[6]=

{K11a322,}

Vassiliev invariants

V2 and V3:

(2, 4)

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 $8$ $32$ $32$ ${\frac {364}{3}}$ ${\frac {68}{3}}$ $256$ ${\frac {1568}{3}}$ ${\frac {128}{3}}$ $128$ ${\frac {256}{3}}$ $512$ ${\frac {2912}{3}}$ ${\frac {544}{3}}$ ${\frac {35911}{15}}$ $-{\frac {3964}{15}}$ ${\frac {55564}{45}}$ ${\frac {905}{9}}$ ${\frac {2551}{15}}$

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 K11a147. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -4 -3 -2 -1 0 1 2 3 4 5 6 7 χ 17                       1 -1 15                     3   3 13                   7 1   -6 11                 9 3     6 9               12 7       -5 7             13 9         4 5           11 12           1 3         10 13             -3 1       6 12               6 -1     3 9                 -6 -3   1 6                   5 -5   3                     -3 -7 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{11}$ ${\mathbb {Z} }^{11}$ $r=2$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{13}$ ${\mathbb {Z} }^{13}$ $r=3$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=4$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=5$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=6$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=7$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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.

K11a146 K11a148