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

K11n99 K11n101

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 X15,22,16,1 X17,20,18,21 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, 9, -11, 8}

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

2t-2 - 11t-1 + 19 - 11t + 2t2

Conway Polynomial:

1 - 3z2 + 2z4

Other knots with the same Alexander/Conway Polynomial:

{937, ...}

Determinant and Signature:

{45, 0}

Jones Polynomial:

- q-3 + 4q-2 - 5q-1 + 7 - 8q + 7q2 - 6q3 + 4q4 - 2q5 + q6

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

{...}

A2 (sl(3)) Invariant:

- q-10 + 2q-8 + 2q-6 - q-4 + 2q-2 - 1 - q6 + q8 - 2q10 + q14 - q16 + q18 + q20

HOMFLY-PT Polynomial:

a-6 - a-4 - 2a-4z2 + a-2z4 + z4 + a2 - a2z2

Kauffman Polynomial:

- a-6 + 4a-6z2 - 4a-6z4 + a-6z6 - 2a-5z + 6a-5z3 - 7a-5z5 + 2a-5z7 - a-4 + 3a-4z2 - 5a-4z6 + 2a-4z8 + 4a-3z3 - 6a-3z5 + a-3z9 - 6a-2z2 + 13a-2z4 - 12a-2z6 + 4a-2z8 + 3a-1z - 4a-1z3 + 2a-1z5 - a-1z7 + a-1z9 - 8z2 + 13z4 - 6z6 + 2z8 + az - az3 + az5 + az7 - a2 - 3a2z2 + 4a2z4 + a3z3

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

{-3, -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 11100. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                   1 j = 11                 1   j = 9               3 1   j = 7             3 1     j = 5           4 3       j = 3         4 3         j = 1       3 4           j = -1     3 5             j = -3   1 2               j = -5   3                 j = -7 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, 100]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 100]]
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, 22, 16, 1], > X[17, 20, 18, 21], X[19, 9, 20, 8], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 100]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -7, 4, 10, -5, -2, 6, -4, 7, -3, -8, 11, -9, 5, -10, > 9, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 100]]
Out[5]=	DTCode[4, 10, 14, 12, -18, 2, 6, -22, -20, -8, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 100]][t]
Out[6]=	2 11 2 19 + -- - -- - 11 t + 2 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 100]][z]
Out[7]=	2 4 1 - 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 37], Knot[11, NonAlternating, 100]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 100]], KnotSignature[Knot[11, NonAlternating, 100]]}
Out[9]=	{45, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 100]][q]
Out[10]=	-3 4 5 2 3 4 5 6 7 - q + -- - - - 8 q + 7 q - 6 q + 4 q - 2 q + q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 100]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 100]][q]
Out[12]=	-10 2 2 -4 2 6 8 10 14 16 18 20 -1 - q + -- + -- - q + -- - q + q - 2 q + q - q + q + q 8 6 2 q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 100]][a, z]
Out[13]=	2 4 -6 -4 2 2 z 2 2 4 z a - a + a - ---- - a z + z + -- 4 2 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 100]][a, z]
Out[14]=	2 2 2 -6 -4 2 2 z 3 z 2 4 z 3 z 6 z 2 2 -a - a - a - --- + --- + a z - 8 z + ---- + ---- - ---- - 3 a z + 5 a 6 4 2 a a a a 3 3 3 4 4 5 6 z 4 z 4 z 3 3 3 4 4 z 13 z 2 4 7 z > ---- + ---- - ---- - a z + a z + 13 z - ---- + ----- + 4 a z - ---- - 5 3 a 6 2 5 a a a a a 5 5 6 6 6 7 7 6 z 2 z 5 6 z 5 z 12 z 2 z z 7 8 > ---- + ---- + a z - 6 z + -- - ---- - ----- + ---- - -- + a z + 2 z + 3 a 6 4 2 5 a a a a a a 8 8 9 9 2 z 4 z z z > ---- + ---- + -- + -- 4 2 3 a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 100]], Vassiliev[3][Knot[11, NonAlternating, 100]]}
Out[15]=	{-3, -3}
In[16]:=	Kh[Knot[11, NonAlternating, 100]][q, t]
Out[16]=	5 1 3 1 2 3 3 3 2 - + 3 q + ----- + ----- + ----- + ---- + --- + 4 q t + 4 q t + 3 q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 5 3 7 3 7 4 9 4 9 5 11 5 13 6 > 4 q t + 3 q t + 3 q t + q t + 3 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n100

K11n99 K11n101