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

K11a28 K11a30

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 16 2 18 22 20 10 6 14

Alexander Polynomial:

2t-3 - 12t-2 + 27t-1 - 33 + 27t - 12t2 + 2t3

Conway Polynomial:

1 - 3z2 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{115, -2}

Jones Polynomial:

q-7 - 3q-6 + 7q-5 - 12q-4 + 16q-3 - 18q-2 + 18q-1 - 16 + 12q - 7q2 + 4q3 - q4

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

{...}

A2 (sl(3)) Invariant:

q-22 - q-18 + 3q-16 - 2q-14 - q-12 + 2q-10 - 3q-8 + 3q-6 - 2q-4 + 1 - 3q2 + 4q4 + 2q10 - q12

HOMFLY-PT Polynomial:

a-2 - a-2z2 - a-2z4 + z2 + 2z4 + z6 - a2z2 + a2z4 + a2z6 - a4 - 3a4z2 - 2a4z4 + a6 + a6z2

Kauffman Polynomial:

2a-3z3 - 3a-3z5 + a-3z7 - a-2 - 3a-2z2 + 16a-2z4 - 15a-2z6 + 4a-2z8 - 2a-1z3 + 13a-1z5 - 16a-1z7 + 5a-1z9 - 11z2 + 36z4 - 33z6 + 5z8 + 2z10 - 4az3 + 18az5 - 29az7 + 11az9 - 10a2z2 + 31a2z4 - 37a2z6 + 10a2z8 + 2a2z10 - 3a3z + 14a3z3 - 15a3z5 - 3a3z7 + 6a3z9 - a4 + 3a4z2 + 4a4z4 - 13a4z6 + 9a4z8 - 3a5z + 12a5z3 - 14a5z5 + 9a5z7 - a6 + 4a6z2 - 6a6z4 + 6a6z6 - 2a7z3 + 3a7z5 - a8z2 + a8z4

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

{-3, 3}

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 1129. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 9                       1 j = 7                     3   j = 5                   4 1   j = 3                 8 3     j = 1               8 4       j = -1             10 8         j = -3           9 9           j = -5         7 9             j = -7       5 9               j = -9     2 7                 j = -11   1 5                   j = -13   2                     j = -15 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, 29]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 29]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[12, 6, 13, 5], X[16, 7, 17, 8], > X[2, 9, 3, 10], X[18, 11, 19, 12], X[22, 14, 1, 13], X[20, 16, 21, 15], > X[10, 17, 11, 18], X[6, 19, 7, 20], X[14, 22, 15, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 29]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -2, 5, -9, 6, -3, 7, -11, 8, -4, 9, -6, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 29]]
Out[5]=	DTCode[4, 8, 12, 16, 2, 18, 22, 20, 10, 6, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 29]][t]
Out[6]=	2 12 27 2 3 -33 + -- - -- + -- + 27 t - 12 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 29]][z]
Out[7]=	2 6 1 - 3 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 29]}
In[9]:=	{KnotDet[Knot[11, Alternating, 29]], KnotSignature[Knot[11, Alternating, 29]]}
Out[9]=	{115, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 29]][q]
Out[10]=	-7 3 7 12 16 18 18 2 3 4 -16 + q - -- + -- - -- + -- - -- + -- + 12 q - 7 q + 4 q - q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 29]}
In[12]:=	A2Invariant[Knot[11, Alternating, 29]][q]
Out[12]=	-22 -18 3 2 -12 2 3 3 2 2 4 10 1 + q - q + --- - --- - q + --- - -- + -- - -- - 3 q + 4 q + 2 q - 16 14 10 8 6 4 q q q q q q 12 > q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 29]][a, z]
Out[13]=	2 4 -2 4 6 2 z 2 2 4 2 6 2 4 z 2 4 a - a + a + z - -- - a z - 3 a z + a z + 2 z - -- + a z - 2 2 a a 4 4 6 2 6 > 2 a z + z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 29]][a, z]
Out[14]=	2 -2 4 6 3 5 2 3 z 2 2 4 2 -a - a - a - 3 a z - 3 a z - 11 z - ---- - 10 a z + 3 a z + 2 a 3 3 6 2 8 2 2 z 2 z 3 3 3 5 3 7 3 > 4 a z - a z + ---- - ---- - 4 a z + 14 a z + 12 a z - 2 a z + 3 a a 4 5 5 4 16 z 2 4 4 4 6 4 8 4 3 z 13 z > 36 z + ----- + 31 a z + 4 a z - 6 a z + a z - ---- + ----- + 2 3 a a a 6 5 3 5 5 5 7 5 6 15 z 2 6 > 18 a z - 15 a z - 14 a z + 3 a z - 33 z - ----- - 37 a z - 2 a 7 7 4 6 6 6 z 16 z 7 3 7 5 7 8 > 13 a z + 6 a z + -- - ----- - 29 a z - 3 a z + 9 a z + 5 z + 3 a a 8 9 4 z 2 8 4 8 5 z 9 3 9 10 2 10 > ---- + 10 a z + 9 a z + ---- + 11 a z + 6 a z + 2 z + 2 a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 29]], Vassiliev[3][Knot[11, Alternating, 29]]}
Out[15]=	{-3, 3}
In[16]:=	Kh[Knot[11, Alternating, 29]][q, t]
Out[16]=	9 10 1 2 1 5 2 7 5 9 -- + -- + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 3 q 15 6 13 5 11 5 11 4 9 4 9 3 7 3 7 2 q q t q t q t q t q t q t q t q t 7 9 9 8 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 8 q t + 4 q t + 8 q t + 3 q t + 4 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a29

K11a28 K11a30