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

K11a96 K11a98

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 18 16 2 20 22 8 6 14

Alexander Polynomial:

2t-3 - 9t-2 + 16t-1 - 17 + 16t - 9t2 + 2t3

Conway Polynomial:

1 - 2z2 + 3z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{71, 2}

Jones Polynomial:

q-5 - 2q-4 + 4q-3 - 7q-2 + 9q-1 - 10 + 11q - 10q2 + 8q3 - 5q4 + 3q5 - q6

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

{...}

A2 (sl(3)) Invariant:

q-16 + q-14 + q-10 - 2q-8 - 2q-6 - q-2 + 3 + q2 + q6 - 2q8 + 2q10 + q16 - q18

HOMFLY-PT Polynomial:

- 2a-4z2 - a-4z4 + 2a-2z2 + 3a-2z4 + a-2z6 + 3 + 3z2 + 3z4 + z6 - 4a2 - 6a2z2 - 2a2z4 + 2a4 + a4z2

Kauffman Polynomial:

a-7z3 - a-6z2 + 3a-6z4 - 3a-5z3 + 5a-5z5 + 2a-4z2 - 9a-4z4 + 7a-4z6 - a-3z + 7a-3z3 - 16a-3z5 + 8a-3z7 - a-2z2 + 8a-2z4 - 17a-2z6 + 7a-2z8 - a-1z + 10a-1z3 - 8a-1z5 - 7a-1z7 + 4a-1z9 + 3 - 19z2 + 46z4 - 36z6 + 6z8 + z10 + 3az - 15az3 + 33az5 - 26az7 + 6az9 + 4a2 - 24a2z2 + 38a2z4 - 18a2z6 + a2z10 + 3a3z - 14a3z3 + 20a3z5 - 11a3z7 + 2a3z9 + 2a4 - 9a4z2 + 12a4z4 - 6a4z6 + a4z8

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 1197. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                       1 j = 11                     2   j = 9                   3 1   j = 7                 5 2     j = 5               5 3       j = 3             6 5         j = 1           5 6           j = -1         4 5             j = -3       3 5               j = -5     1 4                 j = -7   1 3                   j = -9   1                     j = -11 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, 97]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 97]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[18, 7, 19, 8], > X[16, 9, 17, 10], X[2, 11, 3, 12], X[20, 14, 21, 13], X[22, 16, 1, 15], > X[8, 17, 9, 18], X[6, 19, 7, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 97]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -11, 8, -5, 9, -4, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 97]]
Out[5]=	DTCode[4, 10, 12, 18, 16, 2, 20, 22, 8, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 97]][t]
Out[6]=	2 9 16 2 3 -17 + -- - -- + -- + 16 t - 9 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 97]][z]
Out[7]=	2 4 6 1 - 2 z + 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 97]}
In[9]:=	{KnotDet[Knot[11, Alternating, 97]], KnotSignature[Knot[11, Alternating, 97]]}
Out[9]=	{71, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 97]][q]
Out[10]=	-5 2 4 7 9 2 3 4 5 6 -10 + q - -- + -- - -- + - + 11 q - 10 q + 8 q - 5 q + 3 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 97]}
In[12]:=	A2Invariant[Knot[11, Alternating, 97]][q]
Out[12]=	-16 -14 -10 2 2 -2 2 6 8 10 16 18 3 + q + q + q - -- - -- - q + q + q - 2 q + 2 q + q - q 8 6 q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 97]][a, z]
Out[13]=	2 2 4 4 2 4 2 2 z 2 z 2 2 4 2 4 z 3 z 3 - 4 a + 2 a + 3 z - ---- + ---- - 6 a z + a z + 3 z - -- + ---- - 4 2 4 2 a a a a 6 2 4 6 z > 2 a z + z + -- 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 97]][a, z]
Out[14]=	2 2 2 2 4 z z 3 2 z 2 z z 2 2 3 + 4 a + 2 a - -- - - + 3 a z + 3 a z - 19 z - -- + ---- - -- - 24 a z - 3 a 6 4 2 a a a a 3 3 3 3 4 4 2 z 3 z 7 z 10 z 3 3 3 4 3 z > 9 a z + -- - ---- + ---- + ----- - 15 a z - 14 a z + 46 z + ---- - 7 5 3 a 6 a a a a 4 4 5 5 5 9 z 8 z 2 4 4 4 5 z 16 z 8 z 5 > ---- + ---- + 38 a z + 12 a z + ---- - ----- - ---- + 33 a z + 4 2 5 3 a a a a a 6 6 7 7 3 5 6 7 z 17 z 2 6 4 6 8 z 7 z > 20 a z - 36 z + ---- - ----- - 18 a z - 6 a z + ---- - ---- - 4 2 3 a a a a 8 9 7 3 7 8 7 z 4 8 4 z 9 3 9 10 > 26 a z - 11 a z + 6 z + ---- + a z + ---- + 6 a z + 2 a z + z + 2 a a 2 10 > a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 97]], Vassiliev[3][Knot[11, Alternating, 97]]}
Out[15]=	{-2, 2}
In[16]:=	Kh[Knot[11, Alternating, 97]][q, t]
Out[16]=	3 1 1 1 3 1 4 3 5 6 q + 6 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ----- + 11 6 9 5 7 5 7 4 5 4 5 3 3 3 3 2 q t q t q t q t q t q t q t q t 4 5 5 q 3 5 5 2 7 2 7 3 > ---- + --- + --- + 5 q t + 5 q t + 3 q t + 5 q t + 2 q t + 2 q t t q t 9 3 9 4 11 4 13 5 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a97

K11a96 K11a98