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K11a48 K11a50

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X14,6,15,5 X10,8,11,7 X2,9,3,10 X18,12,19,11 X6,14,7,13 X20,16,21,15 X22,18,1,17 X12,20,13,19 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 10 2 18 6 20 22 12 16

Alexander Polynomial:

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

Conway Polynomial:

1 + 2z2 - 5z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{105, 4}

Jones Polynomial:

1 - 2q + 6q2 - 10q3 + 14q4 - 17q5 + 17q6 - 15q7 + 12q8 - 7q9 + 3q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 + q4 + 3q6 - 2q8 + 3q10 - 2q12 - 2q14 + q16 - 4q18 + 3q20 - q22 + 2q24 + 3q26 - 2q28 + q30 - q32 - q34

HOMFLY-PT Polynomial:

- 2a-10 - a-10z2 + 5a-8 + 8a-8z2 + 3a-8z4 - 4a-6 - 8a-6z2 - 7a-6z4 - 2a-6z6 - 2a-4z4 - a-4z6 + 2a-2 + 3a-2z2 + a-2z4

Kauffman Polynomial:

a-13z - 2a-13z3 + a-13z5 + 2a-12z2 - 5a-12z4 + 3a-12z6 + 2a-11z3 - 7a-11z5 + 5a-11z7 + 2a-10 - 8a-10z2 + 10a-10z4 - 10a-10z6 + 6a-10z8 + a-9z + a-9z3 - 2a-9z5 - 2a-9z7 + 4a-9z9 + 5a-8 - 26a-8z2 + 41a-8z4 - 30a-8z6 + 10a-8z8 + a-8z10 + 5a-7z - 13a-7z3 + 16a-7z5 - 14a-7z7 + 7a-7z9 + 4a-6 - 21a-6z2 + 32a-6z4 - 24a-6z6 + 7a-6z8 + a-6z10 + 4a-5z - 8a-5z3 + 5a-5z5 - 5a-5z7 + 3a-5z9 + 2a-4z4 - 6a-4z6 + 3a-4z8 + a-3z + 2a-3z3 - 5a-3z5 + 2a-3z7 - 2a-2 + 5a-2z2 - 4a-2z4 + a-2z6

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

{2, 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 1149. 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                     2   j = 19                   5 1   j = 17                 7 2     j = 15               8 5       j = 13             9 7         j = 11           8 8           j = 9         6 9             j = 7       4 8               j = 5     2 6                 j = 3   1 5                   j = 1   1                     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, 49]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 49]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[14, 6, 15, 5], X[10, 8, 11, 7], > X[2, 9, 3, 10], X[18, 12, 19, 11], X[6, 14, 7, 13], X[20, 16, 21, 15], > X[22, 18, 1, 17], X[12, 20, 13, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 49]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -7, 4, -2, 5, -4, 6, -10, 7, -3, 8, -11, 9, -6, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 49]]
Out[5]=	DTCode[4, 8, 14, 10, 2, 18, 6, 20, 22, 12, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 49]][t]
Out[6]=	3 13 23 2 3 27 - -- + -- - -- - 23 t + 13 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 49]][z]
Out[7]=	2 4 6 1 + 2 z - 5 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 49]}
In[9]:=	{KnotDet[Knot[11, Alternating, 49]], KnotSignature[Knot[11, Alternating, 49]]}
Out[9]=	{105, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 49]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 2 q + 6 q - 10 q + 14 q - 17 q + 17 q - 15 q + 12 q - 7 q + 3 q - 11 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 49]}
In[12]:=	A2Invariant[Knot[11, Alternating, 49]][q]
Out[12]=	4 6 8 10 12 14 16 18 20 22 1 + q + 3 q - 2 q + 3 q - 2 q - 2 q + q - 4 q + 3 q - q + 24 26 28 30 32 34 > 2 q + 3 q - 2 q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 49]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 -2 5 4 2 z 8 z 8 z 3 z 3 z 7 z 2 z z --- + -- - -- + -- - --- + ---- - ---- + ---- + ---- - ---- - ---- + -- - 10 8 6 2 10 8 6 2 8 6 4 2 a a a a a a a a a a a a 6 6 2 z z > ---- - -- 6 4 a a
In[14]:=	Kauffman[Knot[11, Alternating, 49]][a, z]
Out[14]=	2 2 2 2 2 5 4 2 z z 5 z 4 z z 2 z 8 z 26 z 21 z --- + -- + -- - -- + --- + -- + --- + --- + -- + ---- - ---- - ----- - ----- + 10 8 6 2 13 9 7 5 3 12 10 8 6 a a a a a a a a a a a a a 2 3 3 3 3 3 3 4 4 4 5 z 2 z 2 z z 13 z 8 z 2 z 5 z 10 z 41 z > ---- - ---- + ---- + -- - ----- - ---- + ---- - ---- + ----- + ----- + 2 13 11 9 7 5 3 12 10 8 a a a a a a a a a a 4 4 4 5 5 5 5 5 5 6 32 z 2 z 4 z z 7 z 2 z 16 z 5 z 5 z 3 z > ----- + ---- - ---- + --- - ---- - ---- + ----- + ---- - ---- + ---- - 6 4 2 13 11 9 7 5 3 12 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 10 z 30 z 24 z 6 z z 5 z 2 z 14 z 5 z 2 z > ----- - ----- - ----- - ---- + -- + ---- - ---- - ----- - ---- + ---- + 10 8 6 4 2 11 9 7 5 3 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 6 z 10 z 7 z 3 z 4 z 7 z 3 z z z > ---- + ----- + ---- + ---- + ---- + ---- + ---- + --- + --- 10 8 6 4 9 7 5 8 6 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 49]], Vassiliev[3][Knot[11, Alternating, 49]]}
Out[15]=	{2, 6}
In[16]:=	Kh[Knot[11, Alternating, 49]][q, t]
Out[16]=	3 3 5 1 q q 5 7 7 2 9 2 9 3 5 q + 2 q + ---- + - + -- + 6 q t + 4 q t + 8 q t + 6 q t + 9 q t + 2 t t q t 11 3 11 4 13 4 13 5 15 5 15 6 > 8 q t + 8 q t + 9 q t + 7 q t + 8 q t + 5 q t + 17 6 17 7 19 7 19 8 21 8 23 9 > 7 q t + 2 q t + 5 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a49

K11a48 K11a50