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

K11n81 K11n83

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X5,17,6,16 X7,12,8,13 X2,9,3,10 X11,18,12,19 X13,21,14,20 X15,1,16,22 X17,10,18,11 X19,7,20,6 X21,15,22,14

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 3t-2 + 4t-1 - 3 + 4t - 3t2 + t3

Conway Polynomial:

1 + z2 + 3z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{19, 2}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

- q-12 + q-6 + q-4 + 1 + q4 + q6 - q12

HOMFLY-PT Polynomial:

- a-2 - 3a-2z2 - a-2z4 + 3 + 7z2 + 5z4 + z6 - a2 - 3a2z2 - a2z4

Kauffman Polynomial:

a-5z + a-4z2 + 2a-3z - 3a-3z3 + a-3z5 + a-2 - 6a-2z2 + 8a-2z4 - 5a-2z6 + a-2z8 + a-1z - 7a-1z3 + 8a-1z5 - 5a-1z7 + a-1z9 + 3 - 15z2 + 25z4 - 16z6 + 3z8 - az + 2az3 + 2az5 - 4az7 + az9 + a2 - 8a2z2 + 17a2z4 - 11a2z6 + 2a2z8 - a3z + 6a3z3 - 5a3z5 + a3z7

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

{1, 0}

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 1182. 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 j = 9                 1 j = 7               1   j = 5             1 1   j = 3           2 1     j = 1         2 2       j = -1       1 1         j = -3     1 2           j = -5   1 1             j = -7   1               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, NonAlternating, 82]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 82]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[5, 17, 6, 16], X[7, 12, 8, 13], > X[2, 9, 3, 10], X[11, 18, 12, 19], X[13, 21, 14, 20], X[15, 1, 16, 22], > X[17, 10, 18, 11], X[19, 7, 20, 6], X[21, 15, 22, 14]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 82]]
Out[4]=	GaussCode[1, -5, 2, -1, -3, 10, -4, -2, 5, 9, -6, 4, -7, 11, -8, 3, -9, 6, -10, > 7, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 82]]
Out[5]=	DTCode[4, 8, -16, -12, 2, -18, -20, -22, -10, -6, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 82]][t]
Out[6]=	-3 3 4 2 3 -3 + t - -- + - + 4 t - 3 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 82]][z]
Out[7]=	2 4 6 1 + z + 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 82]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 82]], KnotSignature[Knot[11, NonAlternating, 82]]}
Out[9]=	{19, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 82]][q]
Out[10]=	-4 2 2 3 2 3 4 -3 - q + -- - -- + - + 3 q - 2 q + 2 q - q 3 2 q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 82]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 82]][q]
Out[12]=	-12 -6 -4 4 6 12 1 - q + q + q + q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 82]][a, z]
Out[13]=	2 4 -2 2 2 3 z 2 2 4 z 2 4 6 3 - a - a + 7 z - ---- - 3 a z + 5 z - -- - a z + z 2 2 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 82]][a, z]
Out[14]=	2 2 3 -2 2 z 2 z z 3 2 z 6 z 2 2 3 z 3 + a + a + -- + --- + - - a z - a z - 15 z + -- - ---- - 8 a z - ---- - 5 3 a 4 2 3 a a a a a 3 4 5 5 7 z 3 3 3 4 8 z 2 4 z 8 z 5 > ---- + 2 a z + 6 a z + 25 z + ---- + 17 a z + -- + ---- + 2 a z - a 2 3 a a a 6 7 8 3 5 6 5 z 2 6 5 z 7 3 7 8 z > 5 a z - 16 z - ---- - 11 a z - ---- - 4 a z + a z + 3 z + -- + 2 a 2 a a 9 2 8 z 9 > 2 a z + -- + a z a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 82]], Vassiliev[3][Knot[11, NonAlternating, 82]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 82]][q, t]
Out[16]=	3 1 1 1 1 1 2 1 1 2 q 2 q + 2 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + --- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t q t q t 3 5 5 2 7 2 9 3 > q t + q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n82

K11n81 K11n83