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K11n50 K11n52

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,6,15,5 X2837 X9,16,10,17 X11,19,12,18 X6,14,7,13 X15,22,16,1 X17,20,18,21 X19,11,20,10 X21,12,22,13

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 4t-2 - 6t-1 + 7 - 6t + 4t2 - t3

Conway Polynomial:

1 + z2 - 2z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{10127, 10150, ...}

Determinant and Signature:

{29, 0}

Jones Polynomial:

q-2 - 2q-1 + 3 - 4q + 5q2 - 4q3 + 4q4 - 3q5 + 2q6 - q7

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

{...}

A2 (sl(3)) Invariant:

q-6 + q-2 - q2 - q6 + q8 + q10 + q12 + 2q14 - q20 - q22

HOMFLY-PT Polynomial:

- 2a-6 - a-6z2 + 5a-4 + 7a-4z2 + 2a-4z4 - 4a-2 - 8a-2z2 - 5a-2z4 - a-2z6 + 2 + 3z2 + z4

Kauffman Polynomial:

- 3a-7z + 7a-7z3 - 5a-7z5 + a-7z7 + 2a-6 - 7a-6z2 + 14a-6z4 - 10a-6z6 + 2a-6z8 - 5a-5z + 14a-5z3 - 7a-5z5 - 2a-5z7 + a-5z9 + 5a-4 - 17a-4z2 + 27a-4z4 - 19a-4z6 + 4a-4z8 - 3a-3z + 7a-3z3 - 4a-3z5 - 2a-3z7 + a-3z9 + 4a-2 - 15a-2z2 + 16a-2z4 - 9a-2z6 + 2a-2z8 - 2a-1z + 2a-1z3 - 2a-1z5 + a-1z7 + 2 - 4z2 + 3z4 - az + 2az3 + a2z2

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

{1, 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 1151. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 15                   1 j = 13                 1   j = 11               2 1   j = 9             2 1     j = 7           2 2       j = 5         3 2         j = 3       1 2           j = 1     2 3             j = -1   1 2               j = -3   1                 j = -5 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, 51]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 51]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[2, 8, 3, 7], > X[9, 16, 10, 17], X[11, 19, 12, 18], X[6, 14, 7, 13], X[15, 22, 16, 1], > X[17, 20, 18, 21], X[19, 11, 20, 10], X[21, 12, 22, 13]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 51]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, -5, 10, -6, 11, 7, -3, -8, 5, -9, 6, -10, > 9, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 51]]
Out[5]=	DTCode[4, 8, 14, 2, -16, -18, 6, -22, -20, -10, -12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 51]][t]
Out[6]=	-3 4 6 2 3 7 - t + -- - - - 6 t + 4 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 51]][z]
Out[7]=	2 4 6 1 + z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 127], Knot[10, 150], Knot[11, NonAlternating, 51]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 51]], KnotSignature[Knot[11, NonAlternating, 51]]}
Out[9]=	{29, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 51]][q]
Out[10]=	-2 2 2 3 4 5 6 7 3 + q - - - 4 q + 5 q - 4 q + 4 q - 3 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 51]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 51]][q]
Out[12]=	-6 -2 2 6 8 10 12 14 20 22 q + q - q - q + q + q + q + 2 q - q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 51]][a, z]
Out[13]=	2 2 2 4 4 6 2 5 4 2 z 7 z 8 z 4 2 z 5 z z 2 - -- + -- - -- + 3 z - -- + ---- - ---- + z + ---- - ---- - -- 6 4 2 6 4 2 4 2 2 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 51]][a, z]
Out[14]=	2 2 2 2 5 4 3 z 5 z 3 z 2 z 2 7 z 17 z 15 z 2 + -- + -- + -- - --- - --- - --- - --- - a z - 4 z - ---- - ----- - ----- + 6 4 2 7 5 3 a 6 4 2 a a a a a a a a a 3 3 3 3 4 4 2 2 7 z 14 z 7 z 2 z 3 4 14 z 27 z > a z + ---- + ----- + ---- + ---- + 2 a z + 3 z + ----- + ----- + 7 5 3 a 6 4 a a a a a 4 5 5 5 5 6 6 6 7 7 16 z 5 z 7 z 4 z 2 z 10 z 19 z 9 z z 2 z > ----- - ---- - ---- - ---- - ---- - ----- - ----- - ---- + -- - ---- - 2 7 5 3 a 6 4 2 7 5 a a a a a a a a a 7 7 8 8 8 9 9 2 z z 2 z 4 z 2 z z z > ---- + -- + ---- + ---- + ---- + -- + -- 3 a 6 4 2 5 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 51]], Vassiliev[3][Knot[11, NonAlternating, 51]]}
Out[15]=	{1, 3}
In[16]:=	Kh[Knot[11, NonAlternating, 51]][q, t]
Out[16]=	2 1 1 1 3 3 2 5 2 5 3 - + 2 q + ----- + ---- + --- + 3 q t + q t + 2 q t + 3 q t + 2 q t + q 5 2 3 q t q t q t 7 3 7 4 9 4 9 5 11 5 11 6 13 6 15 7 > 2 q t + 2 q t + 2 q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n51

K11n50 K11n52