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

K11a116 K11a118

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 X22,11,1,12 X8,14,9,13 X20,16,21,15 X6,18,7,17 X16,20,17,19 X12,21,13,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 18 2 22 8 20 6 16 12

Alexander Polynomial:

- 2t-3 + 12t-2 - 27t-1 + 35 - 27t + 12t2 - 2t3

Conway Polynomial:

1 + 3z2 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a152, ...}

Determinant and Signature:

{117, 4}

Jones Polynomial:

1 - 3q + 7q2 - 11q3 + 16q4 - 18q5 + 19q6 - 17q7 + 12q8 - 8q9 + 4q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 - q2 + q4 + 2q6 - 3q8 + 4q10 - q12 + 3q16 - 2q18 + 3q20 - 3q22 - q24 + q26 - 3q28 + 2q30 + q32 - q34

HOMFLY-PT Polynomial:

- a-10z2 - a-8 + 2a-8z2 + 2a-8z4 + a-6 + a-6z2 - a-6z4 - a-6z6 - a-4z2 - 2a-4z4 - a-4z6 + a-2 + 2a-2z2 + a-2z4

Kauffman Polynomial:

- a-13z3 + a-13z5 + a-12z2 - 6a-12z4 + 4a-12z6 - a-11z + 6a-11z3 - 12a-11z5 + 7a-11z7 - a-10z2 + 3a-10z4 - 9a-10z6 + 7a-10z8 - a-9z + 14a-9z3 - 18a-9z5 + 4a-9z7 + 4a-9z9 - a-8 - 3a-8z2 + 20a-8z4 - 27a-8z6 + 12a-8z8 + a-8z10 - 3a-7z + 13a-7z3 - 11a-7z5 - 5a-7z7 + 7a-7z9 - a-6 - 2a-6z2 + 14a-6z4 - 22a-6z6 + 9a-6z8 + a-6z10 - 4a-5z + 12a-5z3 - 14a-5z5 + a-5z7 + 3a-5z9 + 2a-4z2 - 7a-4z6 + 4a-4z8 - a-3z + 6a-3z3 - 8a-3z5 + 3a-3z7 - a-2 + 3a-2z2 - 3a-2z4 + a-2z6

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

{3, 6}

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=4 is the signature of 11117. 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 r = 8 r = 9 j = 23                       1 j = 21                     3   j = 19                   5 1   j = 17                 7 3     j = 15               10 5       j = 13             9 7         j = 11           9 10           j = 9         7 9             j = 7       4 9               j = 5     3 7                 j = 3   1 5                   j = 1   2                     j = -1 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, Alternating, 117]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 117]]
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[22, 11, 1, 12], X[8, 14, 9, 13], X[20, 16, 21, 15], > X[6, 18, 7, 17], X[16, 20, 17, 19], X[12, 21, 13, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 117]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -7, 5, -2, 6, -11, 7, -3, 8, -10, 9, -4, 10, > -8, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 117]]
Out[5]=	DTCode[4, 10, 14, 18, 2, 22, 8, 20, 6, 16, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 117]][t]
Out[6]=	2 12 27 2 3 35 - -- + -- - -- - 27 t + 12 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 117]][z]
Out[7]=	2 6 1 + 3 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 117], Knot[11, Alternating, 152]}
In[9]:=	{KnotDet[Knot[11, Alternating, 117]], KnotSignature[Knot[11, Alternating, 117]]}
Out[9]=	{117, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 117]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 3 q + 7 q - 11 q + 16 q - 18 q + 19 q - 17 q + 12 q - 8 q + 4 q - 11 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 117]}
In[12]:=	A2Invariant[Knot[11, Alternating, 117]][q]
Out[12]=	2 4 6 8 10 12 16 18 20 22 24 1 - q + q + 2 q - 3 q + 4 q - q + 3 q - 2 q + 3 q - 3 q - q + 26 28 30 32 34 > q - 3 q + 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 117]][a, z]
Out[13]=	2 2 2 2 2 4 4 4 4 6 6 -8 -6 -2 z 2 z z z 2 z 2 z z 2 z z z z -a + a + a - --- + ---- + -- - -- + ---- + ---- - -- - ---- + -- - -- - -- 10 8 6 4 2 8 6 4 2 6 4 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 117]][a, z]
Out[14]=	2 2 2 2 2 -8 -6 -2 z z 3 z 4 z z z z 3 z 2 z 2 z -a - a - a - --- - -- - --- - --- - -- + --- - --- - ---- - ---- + ---- + 11 9 7 5 3 12 10 8 6 4 a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 3 z z 6 z 14 z 13 z 12 z 6 z 6 z 3 z 20 z > ---- - --- + ---- + ----- + ----- + ----- + ---- - ---- + ---- + ----- + 2 13 11 9 7 5 3 12 10 8 a a a a a a a a a a 4 4 5 5 5 5 5 5 6 6 14 z 3 z z 12 z 18 z 11 z 14 z 8 z 4 z 9 z > ----- - ---- + --- - ----- - ----- - ----- - ----- - ---- + ---- - ---- - 6 2 13 11 9 7 5 3 12 10 a a a a a a a a a a 6 6 6 6 7 7 7 7 7 8 8 27 z 22 z 7 z z 7 z 4 z 5 z z 3 z 7 z 12 z > ----- - ----- - ---- + -- + ---- + ---- - ---- + -- + ---- + ---- + ----- + 8 6 4 2 11 9 7 5 3 10 8 a a a a a a a a a a a 8 8 9 9 9 10 10 9 z 4 z 4 z 7 z 3 z z z > ---- + ---- + ---- + ---- + ---- + --- + --- 6 4 9 7 5 8 6 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 117]], Vassiliev[3][Knot[11, Alternating, 117]]}
Out[15]=	{3, 6}
In[16]:=	Kh[Knot[11, Alternating, 117]][q, t]
Out[16]=	3 3 5 1 2 q q 5 7 7 2 9 2 9 3 5 q + 3 q + ---- + --- + -- + 7 q t + 4 q t + 9 q t + 7 q t + 9 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 9 q t + 10 q t + 9 q t + 7 q t + 10 q t + 5 q t + 17 6 17 7 19 7 19 8 21 8 23 9 > 7 q t + 3 q t + 5 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a117

K11a116 K11a118