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

K11n111 K11n113

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X14,6,15,5 X18,8,19,7 X2,10,3,9 X20,11,21,12 X8,14,9,13 X15,22,16,1 X6,18,7,17 X12,19,13,20 X21,16,22,17

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 5t-2 + 13t-1 - 17 + 13t - 5t2 + t3

Conway Polynomial:

1 + 2z2 + z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{931, K11n11, K11n22, K11n127, ...}

Determinant and Signature:

{55, 2}

Jones Polynomial:

q-1 - 3 + 6q - 8q2 + 10q3 - 9q4 + 8q5 - 6q6 + 3q7 - q8

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

{939, K11n11, ...}

A2 (sl(3)) Invariant:

q-4 - 1 + 2q2 - 2q4 + q6 + q8 + 3q12 - q14 + 2q16 - q18 - 2q20 + q22 - q24

HOMFLY-PT Polynomial:

- 2a-6 - 2a-6z2 - a-6z4 + 4a-4 + 7a-4z2 + 4a-4z4 + a-4z6 - 2a-2 - 4a-2z2 - 2a-2z4 + 1 + z2

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + a-8z2 - 6a-8z4 + 3a-8z6 - 4a-7z + 10a-7z3 - 13a-7z5 + 5a-7z7 + 2a-6 - 7a-6z2 + 11a-6z4 - 10a-6z6 + 4a-6z8 - 6a-5z + 18a-5z3 - 15a-5z5 + 4a-5z7 + a-5z9 + 4a-4 - 15a-4z2 + 24a-4z4 - 15a-4z6 + 5a-4z8 - 2a-3z + 2a-3z3 + 2a-3z5 - a-3z7 + a-3z9 + 2a-2 - 9a-2z2 + 8a-2z4 - 2a-2z6 + a-2z8 - 4a-1z3 + 3a-1z5 + 1 - 2z2 + z4

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

{2, 4}

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 11112. 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 = 17                   1 j = 15                 2   j = 13               4 1   j = 11             4 2     j = 9           5 4       j = 7         5 4         j = 5       3 5           j = 3     3 5             j = 1   1 4               j = -1   2                 j = -3 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, 112]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 112]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[18, 8, 19, 7], > X[2, 10, 3, 9], X[20, 11, 21, 12], X[8, 14, 9, 13], X[15, 22, 16, 1], > X[6, 18, 7, 17], X[12, 19, 13, 20], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 112]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -7, 5, -2, 6, -10, 7, -3, -8, 11, 9, -4, 10, > -6, -11, 8]
In[5]:=	DTCode[Knot[11, NonAlternating, 112]]
Out[5]=	DTCode[4, 10, 14, 18, 2, 20, 8, -22, 6, 12, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 112]][t]
Out[6]=	-3 5 13 2 3 -17 + t - -- + -- + 13 t - 5 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 112]][z]
Out[7]=	2 4 6 1 + 2 z + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 31], Knot[11, NonAlternating, 11], Knot[11, NonAlternating, 22], > Knot[11, NonAlternating, 112], Knot[11, NonAlternating, 127]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 112]], KnotSignature[Knot[11, NonAlternating, 112]]}
Out[9]=	{55, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 112]][q]
Out[10]=	1 2 3 4 5 6 7 8 -3 + - + 6 q - 8 q + 10 q - 9 q + 8 q - 6 q + 3 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[9, 39], Knot[11, NonAlternating, 11], Knot[11, NonAlternating, 112]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 112]][q]
Out[12]=	-4 2 4 6 8 12 14 16 18 20 22 24 -1 + q + 2 q - 2 q + q + q + 3 q - q + 2 q - q - 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 112]][a, z]
Out[13]=	2 2 2 4 4 4 6 2 4 2 2 2 z 7 z 4 z z 4 z 2 z z 1 - -- + -- - -- + z - ---- + ---- - ---- - -- + ---- - ---- + -- 6 4 2 6 4 2 6 4 2 4 a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 112]][a, z]
Out[14]=	2 2 2 2 3 2 4 2 4 z 6 z 2 z 2 z 7 z 15 z 9 z 2 z 1 + -- + -- + -- - --- - --- - --- - 2 z + -- - ---- - ----- - ---- - ---- + 6 4 2 7 5 3 8 6 4 2 9 a a a a a a a a a a a 3 3 3 3 4 4 4 4 5 10 z 18 z 2 z 4 z 4 6 z 11 z 24 z 8 z z > ----- + ----- + ---- - ---- + z - ---- + ----- + ----- + ---- + -- - 7 5 3 a 8 6 4 2 9 a a a a a a a a 5 5 5 5 6 6 6 6 7 7 13 z 15 z 2 z 3 z 3 z 10 z 15 z 2 z 5 z 4 z > ----- - ----- + ---- + ---- + ---- - ----- - ----- - ---- + ---- + ---- - 7 5 3 a 8 6 4 2 7 5 a a a a a a a a a 7 8 8 8 9 9 z 4 z 5 z z z z > -- + ---- + ---- + -- + -- + -- 3 6 4 2 5 3 a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 112]], Vassiliev[3][Knot[11, NonAlternating, 112]]}
Out[15]=	{2, 4}
In[16]:=	Kh[Knot[11, NonAlternating, 112]][q, t]
Out[16]=	3 1 2 q 3 5 5 2 7 2 7 3 4 q + 3 q + ----- + --- + - + 5 q t + 3 q t + 5 q t + 5 q t + 4 q t + 3 2 q t t q t 9 3 9 4 11 4 11 5 13 5 13 6 15 6 > 5 q t + 4 q t + 4 q t + 2 q t + 4 q t + q t + 2 q t + 17 7 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n112

K11n111 K11n113