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K11a93 K11a95

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X12,6,13,5 X18,8,19,7 X2,10,3,9 X8,12,9,11 X20,14,21,13 X22,16,1,15 X6,18,7,17 X16,20,17,19 X14,22,15,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 18 2 8 20 22 6 16 14

Alexander Polynomial:

4t-3 - 13t-2 + 23t-1 - 27 + 23t - 13t2 + 4t3

Conway Polynomial:

1 + 7z2 + 11z4 + 4z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{107, 6}

Jones Polynomial:

q3 - 2q4 + 6q5 - 9q6 + 14q7 - 17q8 + 17q9 - 16q10 + 12q11 - 8q12 + 4q13 - q14

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

{...}

A2 (sl(3)) Invariant:

q10 - q12 + 3q14 + q18 + 4q20 - 2q22 + 3q24 - 2q26 - 2q28 - 4q32 + 2q34 - q36 + 2q40 - q42

HOMFLY-PT Polynomial:

a-12 - a-12z2 - a-12z4 - 4a-10 - 4a-10z2 + a-10z4 + a-10z6 + 2a-8 + 7a-8z2 + 7a-8z4 + 2a-8z6 + 2a-6 + 5a-6z2 + 4a-6z4 + a-6z6

Kauffman Polynomial:

- a-17z3 + a-17z5 + a-16z2 - 6a-16z4 + 4a-16z6 - a-15z + 5a-15z3 - 12a-15z5 + 7a-15z7 + 3a-14z4 - 10a-14z6 + 7a-14z8 - 3a-13z + 10a-13z3 - 11a-13z5 + a-13z7 + 4a-13z9 + a-12 - a-12z2 + 12a-12z4 - 19a-12z6 + 9a-12z8 + a-12z10 - 9a-11z + 17a-11z3 - 7a-11z5 - 6a-11z7 + 6a-11z9 + 4a-10 - 10a-10z2 + 13a-10z4 - 13a-10z6 + 5a-10z8 + a-10z10 - 6a-9z + 15a-9z3 - 14a-9z5 + 2a-9z7 + 2a-9z9 + 2a-8 - 5a-8z2 + 6a-8z4 - 7a-8z6 + 3a-8z8 + a-7z + 2a-7z3 - 5a-7z5 + 2a-7z7 - 2a-6 + 5a-6z2 - 4a-6z4 + a-6z6

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

{7, 17}

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=6 is the signature of 1194. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 r = 10 r = 11 j = 29                       1 j = 27                     3   j = 25                   5 1   j = 23                 7 3     j = 21               9 5       j = 19             8 7         j = 17           9 9           j = 15         5 8             j = 13       4 9               j = 11     2 5                 j = 9     4                   j = 7 1 2                     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, Alternating, 94]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 94]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 6, 13, 5], X[18, 8, 19, 7], > X[2, 10, 3, 9], X[8, 12, 9, 11], X[20, 14, 21, 13], X[22, 16, 1, 15], > X[6, 18, 7, 17], X[16, 20, 17, 19], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 94]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -3, 7, -11, 8, -10, 9, -4, 10, > -7, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 94]]
Out[5]=	DTCode[4, 10, 12, 18, 2, 8, 20, 22, 6, 16, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 94]][t]
Out[6]=	4 13 23 2 3 -27 + -- - -- + -- + 23 t - 13 t + 4 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 94]][z]
Out[7]=	2 4 6 1 + 7 z + 11 z + 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 94]}
In[9]:=	{KnotDet[Knot[11, Alternating, 94]], KnotSignature[Knot[11, Alternating, 94]]}
Out[9]=	{107, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 94]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 2 q + 6 q - 9 q + 14 q - 17 q + 17 q - 16 q + 12 q - 8 q + 13 14 > 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 94]}
In[12]:=	A2Invariant[Knot[11, Alternating, 94]][q]
Out[12]=	10 12 14 18 20 22 24 26 28 32 q - q + 3 q + q + 4 q - 2 q + 3 q - 2 q - 2 q - 4 q + 34 36 40 42 > 2 q - q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 94]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 -12 4 2 2 z 4 z 7 z 5 z z z 7 z 4 z a - --- + -- + -- - --- - ---- + ---- + ---- - --- + --- + ---- + ---- + 10 8 6 12 10 8 6 12 10 8 6 a a a a a a a a a a a 6 6 6 z 2 z z > --- + ---- + -- 10 8 6 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 94]][a, z]
Out[14]=	2 2 2 2 -12 4 2 2 z 3 z 9 z 6 z z z z 10 z 5 z a + --- + -- - -- - --- - --- - --- - --- + -- + --- - --- - ----- - ---- + 10 8 6 15 13 11 9 7 16 12 10 8 a a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 5 z z 5 z 10 z 17 z 15 z 2 z 6 z 3 z 12 z > ---- - --- + ---- + ----- + ----- + ----- + ---- - ---- + ---- + ----- + 6 17 15 13 11 9 7 16 14 12 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 13 z 6 z 4 z z 12 z 11 z 7 z 14 z 5 z 4 z > ----- + ---- - ---- + --- - ----- - ----- - ---- - ----- - ---- + ---- - 10 8 6 17 15 13 11 9 7 16 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 10 z 19 z 13 z 7 z z 7 z z 6 z 2 z 2 z > ----- - ----- - ----- - ---- + -- + ---- + --- - ---- + ---- + ---- + 14 12 10 8 6 15 13 11 9 7 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 7 z 9 z 5 z 3 z 4 z 6 z 2 z z z > ---- + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- 14 12 10 8 13 11 9 12 10 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 94]], Vassiliev[3][Knot[11, Alternating, 94]]}
Out[15]=	{7, 17}
In[16]:=	Kh[Knot[11, Alternating, 94]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 q + q + 2 q t + 4 q t + 2 q t + 5 q t + 4 q t + 9 q t + 15 4 15 5 17 5 17 6 19 6 19 7 > 5 q t + 8 q t + 9 q t + 9 q t + 8 q t + 7 q t + 21 7 21 8 23 8 23 9 25 9 25 10 > 9 q t + 5 q t + 7 q t + 3 q t + 5 q t + q t + 27 10 29 11 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a94

K11a93 K11a95