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

K11a283 K11a285

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

6 10 16 14 20 4 18 2 22 8 12

Alexander Polynomial:

- t-4 + 7t-3 - 21t-2 + 38t-1 - 45 + 38t - 21t2 + 7t3 - t4

Conway Polynomial:

1 + z2 + z4 - z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{179, 2}

Jones Polynomial:

q-3 - 5q-2 + 12q-1 - 19 + 26q - 29q2 + 29q3 - 25q4 + 18q5 - 10q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-8 - 3q-6 + 4q-4 - 2q-2 + 5q2 - 6q4 + 6q6 - 4q8 + 2q10 + 2q12 - 4q14 + 5q16 - 3q18 + q22 - q24

HOMFLY-PT Polynomial:

- a-6 - 2a-6z2 - a-6z4 + 2a-4 + 6a-4z2 + 6a-4z4 + 2a-4z6 - a-2 - 4a-2z2 - 6a-2z4 - 4a-2z6 - a-2z8 + 1 + z2 + 2z4 + z6

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 4a-8z4 + 4a-8z6 - 2a-7z + 8a-7z3 - 12a-7z5 + 9a-7z7 + a-6 - 6a-6z2 + 16a-6z4 - 20a-6z6 + 13a-6z8 - 3a-5z + 12a-5z3 - 9a-5z5 - 8a-5z7 + 11a-5z9 + 2a-4 - 16a-4z2 + 47a-4z4 - 57a-4z6 + 20a-4z8 + 4a-4z10 - a-3z + 3a-3z3 + 11a-3z5 - 38a-3z7 + 22a-3z9 + a-2 - 12a-2z2 + 42a-2z4 - 57a-2z6 + 18a-2z8 + 4a-2z10 + 3a-1z3 - a-1z5 - 16a-1z7 + 11a-1z9 + 1 - 3z2 + 14z4 - 23z6 + 11z8 + 3az3 - 8az5 + 5az7 - a2z4 + a2z6

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

{1, 2}

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 11284. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 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                     3   j = 13                   7 1   j = 11                 11 3     j = 9               14 7       j = 7             15 11         j = 5           14 14           j = 3         12 15             j = 1       8 15               j = -1     4 11                 j = -3   1 8                   j = -5   4                     j = -7 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, 284]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 284]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[16, 6, 17, 5], X[14, 7, 15, 8], > X[20, 10, 21, 9], X[4, 11, 5, 12], X[18, 14, 19, 13], X[2, 16, 3, 15], > X[22, 17, 1, 18], X[8, 20, 9, 19], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 284]]
Out[4]=	GaussCode[1, -8, 2, -6, 3, -1, 4, -10, 5, -2, 6, -11, 7, -4, 8, -3, 9, -7, 10, > -5, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 284]]
Out[5]=	DTCode[6, 10, 16, 14, 20, 4, 18, 2, 22, 8, 12]
In[6]:=	alex = Alexander[Knot[11, Alternating, 284]][t]
Out[6]=	-4 7 21 38 2 3 4 -45 - t + -- - -- + -- + 38 t - 21 t + 7 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 284]][z]
Out[7]=	2 4 6 8 1 + z + z - z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 284]}
In[9]:=	{KnotDet[Knot[11, Alternating, 284]], KnotSignature[Knot[11, Alternating, 284]]}
Out[9]=	{179, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 284]][q]
Out[10]=	-3 5 12 2 3 4 5 6 7 8 -19 + q - -- + -- + 26 q - 29 q + 29 q - 25 q + 18 q - 10 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 284]}
In[12]:=	A2Invariant[Knot[11, Alternating, 284]][q]
Out[12]=	-8 3 4 2 2 4 6 8 10 12 14 q - -- + -- - -- + 5 q - 6 q + 6 q - 4 q + 2 q + 2 q - 4 q + 6 4 2 q q q 16 18 22 24 > 5 q - 3 q + q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 284]][a, z]
Out[13]=	2 2 2 4 4 4 -6 2 -2 2 2 z 6 z 4 z 4 z 6 z 6 z 6 1 - a + -- - a + z - ---- + ---- - ---- + 2 z - -- + ---- - ---- + z + 4 6 4 2 6 4 2 a a a a a a a 6 6 8 2 z 4 z z > ---- - ---- - -- 4 2 2 a a a
In[14]:=	Kauffman[Knot[11, Alternating, 284]][a, z]
Out[14]=	2 2 2 2 3 -6 2 -2 2 z 3 z z 2 z 6 z 16 z 12 z z 1 + a + -- + a - --- - --- - -- - 3 z + -- - ---- - ----- - ----- - -- + 4 7 5 3 8 6 4 2 9 a a a a a a a a a 3 3 3 3 4 4 4 8 z 12 z 3 z 3 z 3 4 4 z 16 z 47 z > ---- + ----- + ---- + ---- + 3 a z + 14 z - ---- + ----- + ----- + 7 5 3 a 8 6 4 a a a a a a 4 5 5 5 5 5 6 42 z 2 4 z 12 z 9 z 11 z z 5 6 4 z > ----- - a z + -- - ----- - ---- + ----- - -- - 8 a z - 23 z + ---- - 2 9 7 5 3 a 8 a a a a a a 6 6 6 7 7 7 7 20 z 57 z 57 z 2 6 9 z 8 z 38 z 16 z 7 > ----- - ----- - ----- + a z + ---- - ---- - ----- - ----- + 5 a z + 6 4 2 7 5 3 a a a a a a a 8 8 8 9 9 9 10 10 8 13 z 20 z 18 z 11 z 22 z 11 z 4 z 4 z > 11 z + ----- + ----- + ----- + ----- + ----- + ----- + ----- + ----- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 284]], Vassiliev[3][Knot[11, Alternating, 284]]}
Out[15]=	{1, 2}
In[16]:=	Kh[Knot[11, Alternating, 284]][q, t]
Out[16]=	3 1 4 1 8 4 11 8 q 3 15 q + 12 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 15 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 14 q t + 14 q t + 15 q t + 11 q t + 14 q t + 7 q t + 11 q t + 11 5 13 5 13 6 15 6 17 7 > 3 q t + 7 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a284

K11a283 K11a285