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

K11n96 K11n98

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X14,6,15,5 X12,8,13,7 X9,19,10,18 X2,11,3,12 X6,14,7,13 X15,20,16,21 X17,22,18,1 X19,9,20,8 X21,16,22,17

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 12 -18 2 6 -20 -22 -8 -16

Alexander Polynomial:

- 2t-1 + 5 - 2t

Conway Polynomial:

1 - 2z2

Other knots with the same Alexander/Conway Polynomial:

{61, 946, K11n67, K11n139, ...}

Determinant and Signature:

{9, 0}

Jones Polynomial:

- q-5 + 2q-4 - 2q-3 + 3q-2 - 2q-1 + 2 - q - q2 + q3 - q4 + q5

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

{...}

A2 (sl(3)) Invariant:

- q-16 + 2q-8 + q-6 + 2q-4 + q-2 - 3q4 - q6 - q8 + q12 + q14 + q16

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

2a-4 - 11a-4z2 + 15a-4z4 - 7a-4z6 + a-4z8 + 5a-3z - 13a-3z3 + 15a-3z5 - 7a-3z7 + a-3z9 + 3a-2 - 22a-2z2 + 32a-2z4 - 15a-2z6 + 2a-2z8 + 8a-1z - 18a-1z3 + 19a-1z5 - 8a-1z7 + a-1z9 - 9z2 + 17z4 - 8z6 + z8 + 4az - 4az3 + az5 - 3a2 + 7a2z2 - 7a2z4 + 2a2z6 + 2a3z - 2a3z3 - 2a3z5 + a3z7 - a4 + 5a4z2 - 7a4z4 + 2a4z6 + a5z - 3a5z3 + a5z5

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

{-2, -3}

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=0 is the signature of 1197. 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 = 11                       1 j = 9                         j = 7                   1 1   j = 5               1 1       j = 3             1   1       j = 1           2 2 1         j = -1         2 2             j = -3       1 1 1             j = -5     1 2                 j = -7   1 1                   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, NonAlternating, 97]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 97]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 6, 15, 5], X[12, 8, 13, 7], > X[9, 19, 10, 18], X[2, 11, 3, 12], X[6, 14, 7, 13], X[15, 20, 16, 21], > X[17, 22, 18, 1], X[19, 9, 20, 8], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 97]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -7, 4, 10, -5, -2, 6, -4, 7, -3, -8, 11, -9, 5, -10, > 8, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 97]]
Out[5]=	DTCode[4, 10, 14, 12, -18, 2, 6, -20, -22, -8, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 97]][t]
Out[6]=	2 5 - - - 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 97]][z]
Out[7]=	2 1 - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[6, 1], Knot[9, 46], Knot[11, NonAlternating, 67], > Knot[11, NonAlternating, 97], Knot[11, NonAlternating, 139]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 97]], KnotSignature[Knot[11, NonAlternating, 97]]}
Out[9]=	{9, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 97]][q]
Out[10]=	-5 2 2 3 2 2 3 4 5 2 - q + -- - -- + -- - - - q - q + q - 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, NonAlternating, 97]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 97]][q]
Out[12]=	-16 2 -6 2 -2 4 6 8 12 14 16 -q + -- + q + -- + q - 3 q - q - q + q + q + q 8 4 q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 97]][a, z]
Out[13]=	2 2 4 2 3 2 4 2 z 4 z 2 2 4 2 z 2 4 -- - -- + 3 a - a - z + -- - ---- + 3 a z - a z - -- + a z 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 97]][a, z]
Out[14]=	2 2 3 2 4 5 z 8 z 3 5 2 11 z -- + -- - 3 a - a + --- + --- + 4 a z + 2 a z + a z - 9 z - ----- - 4 2 3 a 4 a a a a 2 3 3 22 z 2 2 4 2 13 z 18 z 3 3 3 5 3 > ----- + 7 a z + 5 a z - ----- - ----- - 4 a z - 2 a z - 3 a z + 2 3 a a a 4 4 5 5 4 15 z 32 z 2 4 4 4 15 z 19 z 5 > 17 z + ----- + ----- - 7 a z - 7 a z + ----- + ----- + a z - 4 2 3 a a a a 6 6 7 7 3 5 5 5 6 7 z 15 z 2 6 4 6 7 z 8 z > 2 a z + a z - 8 z - ---- - ----- + 2 a z + 2 a z - ---- - ---- + 4 2 3 a a a a 8 8 9 9 3 7 8 z 2 z z z > a z + z + -- + ---- + -- + -- 4 2 3 a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 97]], Vassiliev[3][Knot[11, NonAlternating, 97]]}
Out[15]=	{-2, -3}
In[16]:=	Kh[Knot[11, NonAlternating, 97]][q, t]
Out[16]=	-3 2 1 1 1 1 1 2 1 1 q + - + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 2 3 2 5 2 3 3 5 3 7 4 7 5 11 6 > --- + 2 q t + q t + q t + q t + q t + q t + q t + q t + q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n97

K11n96 K11n98