© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11a98 K11a100

Knotscape

This page is passe. Go here instead!    The Knot K11a99 Visit K11a99's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X12,6,13,5 X20,8,21,7 X16,10,17,9 X2,11,3,12 X18,13,19,14 X8,16,9,15 X22,17,1,18 X6,20,7,19 X14,21,15,22

Gauss Code:

{1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -11, 8, -5, 9, -7, 10, -4, 11, -9}

DT (Dowker-Thistlethwaite) Code:

4 10 12 20 16 2 18 8 22 6 14

Alexander Polynomial:

- t-4 + 6t-3 - 17t-2 + 28t-1 - 31 + 28t - 17t2 + 6t3 - t4

Conway Polynomial:

1 - 2z2 - z4 - 2z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{K11a277, ...}

Determinant and Signature:

{135, 2}

Jones Polynomial:

- q-4 + 4q-3 - 8q-2 + 14q-1 - 18 + 21q - 22q2 + 19q3 - 14q4 + 9q5 - 4q6 + q7

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-12 + q-10 + q-8 - q-6 + 4q-4 - 2q-2 + 2 + q2 - 4q4 + 3q6 - 5q8 + 3q10 - q14 + 3q16 - 2q18 + q20

HOMFLY-PT Polynomial:

2a-4 + 3a-4z2 + 3a-4z4 + a-4z6 - 4a-2 - 10a-2z2 - 10a-2z4 - 5a-2z6 - a-2z8 + 3 + 7z2 + 7z4 + 2z6 - 2a2z2 - a2z4

Kauffman Polynomial:

a-8z4 - a-7z3 + 4a-7z5 + 3a-6z2 - 8a-6z4 + 9a-6z6 - 2a-5z + 9a-5z3 - 17a-5z5 + 13a-5z7 + 2a-4 - 5a-4z2 + 7a-4z4 - 18a-4z6 + 13a-4z8 - 3a-3z + 18a-3z3 - 25a-3z5 - a-3z7 + 8a-3z9 + 4a-2 - 23a-2z2 + 55a-2z4 - 62a-2z6 + 19a-2z8 + 2a-2z10 - 3a-1z + 15a-1z3 - 27a-1z7 + 13a-1z9 + 3 - 21z2 + 55z4 - 49z6 + 10z8 + 2z10 - 3az + 10az3 + az5 - 12az7 + 5az9 - 6a2z2 + 16a2z4 - 14a2z6 + 4a2z8 - a3z + 3a3z3 - 3a3z5 + a3z7

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-2, -2}

Khovanov Homology: (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 1199. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 15                       1 j = 13                     3   j = 11                   6 1   j = 9                 8 3     j = 7               11 6       j = 5             11 8         j = 3           10 11           j = 1         9 12             j = -1       5 9               j = -3     3 9                 j = -5   1 5                   j = -7   3                     j = -9 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, Alternating, 99]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 99]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[20, 8, 21, 7], > X[16, 10, 17, 9], X[2, 11, 3, 12], X[18, 13, 19, 14], X[8, 16, 9, 15], > X[22, 17, 1, 18], X[6, 20, 7, 19], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 99]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -3, 7, -11, 8, -5, 9, -7, 10, > -4, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 99]]
Out[5]=	DTCode[4, 10, 12, 20, 16, 2, 18, 8, 22, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 99]][t]
Out[6]=	-4 6 17 28 2 3 4 -31 - t + -- - -- + -- + 28 t - 17 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 99]][z]
Out[7]=	2 4 6 8 1 - 2 z - z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 99], Knot[11, Alternating, 277]}
In[9]:=	{KnotDet[Knot[11, Alternating, 99]], KnotSignature[Knot[11, Alternating, 99]]}
Out[9]=	{135, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 99]][q]
Out[10]=	-4 4 8 14 2 3 4 5 6 7 -18 - q + -- - -- + -- + 21 q - 22 q + 19 q - 14 q + 9 q - 4 q + q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 99]}
In[12]:=	A2Invariant[Knot[11, Alternating, 99]][q]
Out[12]=	-12 -10 -8 -6 4 2 2 4 6 8 10 14 2 - q + q + q - q + -- - -- + q - 4 q + 3 q - 5 q + 3 q - q + 4 2 q q 16 18 20 > 3 q - 2 q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 99]][a, z]
Out[13]=	2 2 4 4 2 4 2 3 z 10 z 2 2 4 3 z 10 z 2 4 3 + -- - -- + 7 z + ---- - ----- - 2 a z + 7 z + ---- - ----- - a z + 4 2 4 2 4 2 a a a a a a 6 6 8 6 z 5 z z > 2 z + -- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 99]][a, z]
Out[14]=	2 2 2 2 4 2 z 3 z 3 z 3 2 3 z 5 z 23 z 3 + -- + -- - --- - --- - --- - 3 a z - a z - 21 z + ---- - ---- - ----- - 4 2 5 3 a 6 4 2 a a a a a a a 3 3 3 3 4 2 2 z 9 z 18 z 15 z 3 3 3 4 z > 6 a z - -- + ---- + ----- + ----- + 10 a z + 3 a z + 55 z + -- - 7 5 3 a 8 a a a a 4 4 4 5 5 5 8 z 7 z 55 z 2 4 4 z 17 z 25 z 5 3 5 > ---- + ---- + ----- + 16 a z + ---- - ----- - ----- + a z - 3 a z - 6 4 2 7 5 3 a a a a a a 6 6 6 7 7 7 6 9 z 18 z 62 z 2 6 13 z z 27 z 7 > 49 z + ---- - ----- - ----- - 14 a z + ----- - -- - ----- - 12 a z + 6 4 2 5 3 a a a a a a 8 8 9 9 3 7 8 13 z 19 z 2 8 8 z 13 z 9 10 > a z + 10 z + ----- + ----- + 4 a z + ---- + ----- + 5 a z + 2 z + 4 2 3 a a a a 10 2 z > ----- 2 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 99]], Vassiliev[3][Knot[11, Alternating, 99]]}
Out[15]=	{-2, -2}
In[16]:=	Kh[Knot[11, Alternating, 99]][q, t]
Out[16]=	3 1 3 1 5 3 9 5 9 12 q + 10 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t q t 9 q 3 5 5 2 7 2 7 3 9 3 > --- + 11 q t + 11 q t + 8 q t + 11 q t + 6 q t + 8 q t + t 9 4 11 4 11 5 13 5 15 6 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a99

K11a98 K11a100