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

K11n114 K11n116

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X14,6,15,5 X7,19,8,18 X9,17,10,16 X2,11,3,12 X20,13,21,14 X22,16,1,15 X17,9,18,8 X12,19,13,20 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 6t-2 - 18t-1 + 27 - 18t + 6t2 - t3

Conway Polynomial:

1 - 3z2 - z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{77, 0}

Jones Polynomial:

- q-5 + 4q-4 - 7q-3 + 11q-2 - 13q-1 + 13 - 12q + 9q2 - 5q3 + 2q4

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

{...}

A2 (sl(3)) Invariant:

- q-16 + q-14 + 2q-12 - 2q-10 + 3q-8 - q-4 + 2q-2 - 3 + 2q2 - 3q4 + 2q8 - 2q10 + 2q12 + q14

HOMFLY-PT Polynomial:

a-4 + a-2z4 - 2 - 5z2 - 3z4 - z6 + 2a2 + 3a2z2 + 2a2z4 - a4z2

Kauffman Polynomial:

a-4 - 4a-4z2 + 3a-4z4 + a-3z - 4a-3z3 + 3a-3z5 + a-3z7 - a-2z2 - a-2z4 + a-2z6 + 2a-2z8 - 2a-1z + 6a-1z3 - 10a-1z5 + 6a-1z7 + a-1z9 - 2 + 12z2 - 14z4 - z6 + 6z8 - 4az + 15az3 - 24az5 + 11az7 + az9 - 2a2 + 12a2z2 - 17a2z4 + 2a2z6 + 4a2z8 - a3z + 4a3z3 - 10a3z5 + 6a3z7 + 3a4z2 - 7a4z4 + 4a4z6 - a5z3 + a5z5

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

{-3, -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 11115. 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 j = 9                   2 j = 7                 3   j = 5               6 2   j = 3             6 3     j = 1           7 6       j = -1         7 7         j = -3       4 6           j = -5     3 7             j = -7   1 4               j = -9   3                 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, 115]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 115]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 6, 15, 5], X[7, 19, 8, 18], > X[9, 17, 10, 16], X[2, 11, 3, 12], X[20, 13, 21, 14], X[22, 16, 1, 15], > X[17, 9, 18, 8], X[12, 19, 13, 20], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 115]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, -4, 9, -5, -2, 6, -10, 7, -3, 8, 5, -9, 4, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 115]]
Out[5]=	DTCode[4, 10, 14, -18, -16, 2, 20, 22, -8, 12, 6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 115]][t]
Out[6]=	-3 6 18 2 3 27 - t + -- - -- - 18 t + 6 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 115]][z]
Out[7]=	2 6 1 - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 115]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 115]], KnotSignature[Knot[11, NonAlternating, 115]]}
Out[9]=	{77, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 115]][q]
Out[10]=	-5 4 7 11 13 2 3 4 13 - q + -- - -- + -- - -- - 12 q + 9 q - 5 q + 2 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, 115]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 115]][q]
Out[12]=	-16 -14 2 2 3 -4 2 2 4 8 10 -3 - q + q + --- - --- + -- - q + -- + 2 q - 3 q + 2 q - 2 q + 12 10 8 2 q q q q 12 14 > 2 q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 115]][a, z]
Out[13]=	4 -4 2 2 2 2 4 2 4 z 2 4 6 -2 + a + 2 a - 5 z + 3 a z - a z - 3 z + -- + 2 a z - z 2 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 115]][a, z]
Out[14]=	2 2 -4 2 z 2 z 3 2 4 z z 2 2 -2 + a - 2 a + -- - --- - 4 a z - a z + 12 z - ---- - -- + 12 a z + 3 a 4 2 a a a 3 3 4 4 4 2 4 z 6 z 3 3 3 5 3 4 3 z z > 3 a z - ---- + ---- + 15 a z + 4 a z - a z - 14 z + ---- - -- - 3 a 4 2 a a a 5 5 6 2 4 4 4 3 z 10 z 5 3 5 5 5 6 z > 17 a z - 7 a z + ---- - ----- - 24 a z - 10 a z + a z - z + -- + 3 a 2 a a 7 7 8 2 6 4 6 z 6 z 7 3 7 8 2 z 2 8 > 2 a z + 4 a z + -- + ---- + 11 a z + 6 a z + 6 z + ---- + 4 a z + 3 a 2 a a 9 z 9 > -- + a z a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 115]], Vassiliev[3][Knot[11, NonAlternating, 115]]}
Out[15]=	{-3, -1}
In[16]:=	Kh[Knot[11, NonAlternating, 115]][q, t]
Out[16]=	7 1 3 1 4 3 7 4 6 7 - + 7 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 q t 3 3 2 5 2 5 3 7 3 9 4 > 6 q t + 6 q t + 3 q t + 6 q t + 2 q t + 3 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n115

K11n114 K11n116