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

K11n142 K11n144

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X12,4,13,3 X5,16,6,17 X22,8,1,7 X9,18,10,19 X2,12,3,11 X13,21,14,20 X15,10,16,11 X17,6,18,7 X19,15,20,14 X8,22,9,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

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

Conway Polynomial:

1 - 2z2 - 4z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{9, 0}

Jones Polynomial:

q-3 - q-2 + 1 - q + 2q2 - 2q3 + 2q4 - 2q5 + q6

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

{...}

A2 (sl(3)) Invariant:

q-10 + q-6 - q2 + q4 + q8 - q12 + q18

HOMFLY-PT Polynomial:

a-4 + 3a-4z2 + a-4z4 - a-2 - 6a-2z2 - 5a-2z4 - a-2z6 + a2 + a2z2

Kauffman Polynomial:

3a-6z2 - 4a-6z4 + a-6z6 - 2a-5z + 9a-5z3 - 9a-5z5 + 2a-5z7 + a-4 - 2a-4z2 + 3a-4z4 - 4a-4z6 + a-4z8 - 4a-3z + 12a-3z3 - 10a-3z5 + 2a-3z7 + a-2 - 7a-2z2 + 10a-2z4 - 6a-2z6 + a-2z8 - 4a-1z + 11a-1z3 - 7a-1z5 + a-1z7 + 3z2 - 2z4 - 2az + 8az3 - 6az5 + az7 - a2 + 5a2z2 - 5a2z4 + a2z6

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

{-2, -1}

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 11143. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 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                 1 1   j = 7             1 2 1     j = 5             1 1       j = 3         1 2 2         j = 1       1 2 1           j = -1       1 2             j = -3   1 1 1               j = -5                       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, 143]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 143]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[5, 16, 6, 17], X[22, 8, 1, 7], > X[9, 18, 10, 19], X[2, 12, 3, 11], X[13, 21, 14, 20], X[15, 10, 16, 11], > X[17, 6, 18, 7], X[19, 15, 20, 14], X[8, 22, 9, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 143]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 9, 4, -11, -5, 8, 6, -2, -7, 10, -8, 3, -9, 5, -10, > 7, 11, -4]
In[5]:=	DTCode[Knot[11, NonAlternating, 143]]
Out[5]=	DTCode[4, 12, -16, 22, -18, 2, -20, -10, -6, -14, 8]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 143]][t]
Out[6]=	-3 2 1 2 3 1 - t + -- - - - t + 2 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 143]][z]
Out[7]=	2 4 6 1 - 2 z - 4 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 143]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 143]], KnotSignature[Knot[11, NonAlternating, 143]]}
Out[9]=	{9, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 143]][q]
Out[10]=	-3 -2 2 3 4 5 6 1 + q - q - q + 2 q - 2 q + 2 q - 2 q + q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 143]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 143]][q]
Out[12]=	-10 -6 2 4 8 12 18 q + q - q + q + q - q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 143]][a, z]
Out[13]=	2 2 4 4 6 -4 -2 2 3 z 6 z 2 2 z 5 z z a - a + a + ---- - ---- + a z + -- - ---- - -- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 143]][a, z]
Out[14]=	2 2 2 -4 -2 2 2 z 4 z 4 z 2 3 z 2 z 7 z a + a - a - --- - --- - --- - 2 a z + 3 z + ---- - ---- - ---- + 5 3 a 6 4 2 a a a a a 3 3 3 4 4 4 2 2 9 z 12 z 11 z 3 4 4 z 3 z 10 z > 5 a z + ---- + ----- + ----- + 8 a z - 2 z - ---- + ---- + ----- - 5 3 a 6 4 2 a a a a a 5 5 5 6 6 6 7 2 4 9 z 10 z 7 z 5 z 4 z 6 z 2 6 2 z > 5 a z - ---- - ----- - ---- - 6 a z + -- - ---- - ---- + a z + ---- + 5 3 a 6 4 2 5 a a a a a a 7 7 8 8 2 z z 7 z z > ---- + -- + a z + -- + -- 3 a 4 2 a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 143]], Vassiliev[3][Knot[11, NonAlternating, 143]]}
Out[15]=	{-2, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 143]][q, t]
Out[16]=	2 3 1 1 1 1 1 q 3 - + 2 q + q + ----- + ----- + ----- + ---- + --- + - + q t + 2 q t + q 7 4 3 3 3 2 3 q t t q t q t q t q t 3 2 5 2 7 2 5 3 7 3 7 4 9 4 9 5 > 2 q t + q t + q t + q t + 2 q t + q t + q t + q t + 11 5 13 6 > q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n143

K11n142 K11n144