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

K11a274 K11a276

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,4,11,3 X14,6,15,5 X20,8,21,7 X2,10,3,9 X18,12,19,11 X4,14,5,13 X22,15,1,16 X12,18,13,17 X8,20,9,19 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 14 20 2 18 4 22 12 8 16

Alexander Polynomial:

- 3t-3 + 15t-2 - 29t-1 + 35 - 29t + 15t2 - 3t3

Conway Polynomial:

1 + 4z2 - 3z4 - 3z6

Other knots with the same Alexander/Conway Polynomial:

{K11a344, ...}

Determinant and Signature:

{129, 4}

Jones Polynomial:

1 - 3q + 8q2 - 13q3 + 18q4 - 21q5 + 21q6 - 18q7 + 14q8 - 8q9 + 3q10 - q11

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

{...}

A2 (sl(3)) Invariant:

1 - q2 + q4 + 3q6 - 3q8 + 4q10 - 2q12 - q14 + 2q16 - 4q18 + 4q20 - 2q22 + 2q24 + 3q26 - 3q28 + q30 - q32 - q34

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

a-13z - 2a-13z3 + a-13z5 + a-12z2 - 4a-12z4 + 3a-12z6 - 2a-11z + 6a-11z3 - 9a-11z5 + 6a-11z7 + 2a-10 - 10a-10z2 + 15a-10z4 - 14a-10z6 + 8a-10z8 - 2a-9z + 8a-9z3 - 5a-9z5 - 4a-9z7 + 6a-9z9 + 4a-8 - 20a-8z2 + 36a-8z4 - 32a-8z6 + 11a-8z8 + 2a-8z10 + 2a-7z - 6a-7z3 + 13a-7z5 - 20a-7z7 + 11a-7z9 + 3a-6 - 14a-6z2 + 25a-6z4 - 26a-6z6 + 8a-6z8 + 2a-6z10 + a-5z - 2a-5z3 + a-5z5 - 7a-5z7 + 5a-5z9 + a-4 - 2a-4z2 + 5a-4z4 - 10a-4z6 + 5a-4z8 + 4a-3z3 - 7a-3z5 + 3a-3z7 - a-2 + 3a-2z2 - 3a-2z4 + a-2z6

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

{4, 11}

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 11275. 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                   6 1   j = 17                 8 2     j = 15               10 6       j = 13             11 8         j = 11           10 10           j = 9         8 11             j = 7       5 10               j = 5     3 8                 j = 3   1 6                   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, 275]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 275]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 8, 21, 7], > X[2, 10, 3, 9], X[18, 12, 19, 11], X[4, 14, 5, 13], X[22, 15, 1, 16], > X[12, 18, 13, 17], X[8, 20, 9, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 275]]
Out[4]=	GaussCode[1, -5, 2, -7, 3, -1, 4, -10, 5, -2, 6, -9, 7, -3, 8, -11, 9, -6, 10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 275]]
Out[5]=	DTCode[6, 10, 14, 20, 2, 18, 4, 22, 12, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 275]][t]
Out[6]=	3 15 29 2 3 35 - -- + -- - -- - 29 t + 15 t - 3 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 275]][z]
Out[7]=	2 4 6 1 + 4 z - 3 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 275], Knot[11, Alternating, 344]}
In[9]:=	{KnotDet[Knot[11, Alternating, 275]], KnotSignature[Knot[11, Alternating, 275]]}
Out[9]=	{129, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 275]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 1 - 3 q + 8 q - 13 q + 18 q - 21 q + 21 q - 18 q + 14 q - 8 q + 3 q - 11 > q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 275]}
In[12]:=	A2Invariant[Knot[11, Alternating, 275]][q]
Out[12]=	2 4 6 8 10 12 14 16 18 20 1 - q + q + 3 q - 3 q + 4 q - 2 q - q + 2 q - 4 q + 4 q - 22 24 26 28 30 32 34 > 2 q + 2 q + 3 q - 3 q + q - q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 275]][a, z]
Out[13]=	2 2 2 2 2 4 4 -2 4 3 -4 -2 z 7 z 6 z 2 z 2 z 3 z 6 z --- + -- - -- + a + a - --- + ---- - ---- + ---- + ---- + ---- - ---- - 10 8 6 10 8 6 4 2 8 6 a a a a a a a a a a 4 4 6 6 z z 2 z z > -- + -- - ---- - -- 4 2 6 4 a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 275]][a, z]
Out[14]=	2 2 2 2 4 3 -4 -2 z 2 z 2 z 2 z z z 10 z 20 z --- + -- + -- + a - a + --- - --- - --- + --- + -- + --- - ----- - ----- - 10 8 6 13 11 9 7 5 12 10 8 a a a a a a a a a a a 2 2 2 3 3 3 3 3 3 4 14 z 2 z 3 z 2 z 6 z 8 z 6 z 2 z 4 z 4 z > ----- - ---- + ---- - ---- + ---- + ---- - ---- - ---- + ---- - ---- + 6 4 2 13 11 9 7 5 3 12 a a a a a a a a a a 4 4 4 4 4 5 5 5 5 5 15 z 36 z 25 z 5 z 3 z z 9 z 5 z 13 z z > ----- + ----- + ----- + ---- - ---- + --- - ---- - ---- + ----- + -- - 10 8 6 4 2 13 11 9 7 5 a a a a a a a a a a 5 6 6 6 6 6 6 7 7 7 7 z 3 z 14 z 32 z 26 z 10 z z 6 z 4 z 20 z > ---- + ---- - ----- - ----- - ----- - ----- + -- + ---- - ---- - ----- - 3 12 10 8 6 4 2 11 9 7 a a a a a a a a a a 7 7 8 8 8 8 9 9 9 10 7 z 3 z 8 z 11 z 8 z 5 z 6 z 11 z 5 z 2 z > ---- + ---- + ---- + ----- + ---- + ---- + ---- + ----- + ---- + ----- + 5 3 10 8 6 4 9 7 5 8 a a a a a a a a a a 10 2 z > ----- 6 a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 275]], Vassiliev[3][Knot[11, Alternating, 275]]}
Out[15]=	{4, 11}
In[16]:=	Kh[Knot[11, Alternating, 275]][q, t]
Out[16]=	3 3 5 1 2 q q 5 7 7 2 9 2 6 q + 3 q + ---- + --- + -- + 8 q t + 5 q t + 10 q t + 8 q t + 2 t t q t 9 3 11 3 11 4 13 4 13 5 15 5 > 11 q t + 10 q t + 10 q t + 11 q t + 8 q t + 10 q t + 15 6 17 6 17 7 19 7 19 8 21 8 23 9 > 6 q t + 8 q t + 2 q t + 6 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a275

K11a274 K11a276