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

K11a81 K11a83

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 16 2 22 18 20 6 14 8

Alexander Polynomial:

- t-4 + 5t-3 - 12t-2 + 19t-1 - 21 + 19t - 12t2 + 5t3 - t4

Conway Polynomial:

1 - 2z4 - 3z6 - z8

Other knots with the same Alexander/Conway Polynomial:

{10116, K11a7, K11a33, ...}

Determinant and Signature:

{95, -2}

Jones Polynomial:

q-7 - 3q-6 + 6q-5 - 10q-4 + 13q-3 - 15q-2 + 15q-1 - 12 + 10q - 6q2 + 3q3 - q4

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

{K11a33, ...}

A2 (sl(3)) Invariant:

q-20 - q-18 + 2q-16 - q-14 - q-12 + q-10 - 4q-8 + 2q-6 - 2q-4 + 2q-2 + 3 + 3q4 - q6 - q12

HOMFLY-PT Polynomial:

- 2a-2 - 3a-2z2 - a-2z4 + 7 + 13z2 + 9z4 + 2z6 - 6a2 - 15a2z2 - 14a2z4 - 6a2z6 - a2z8 + 2a4 + 5a4z2 + 4a4z4 + a4z6

Kauffman Polynomial:

- 2a-3z + 5a-3z3 - 4a-3z5 + a-3z7 + 2a-2 - 8a-2z2 + 15a-2z4 - 12a-2z6 + 3a-2z8 - 6a-1z + 13a-1z3 - 4a-1z5 - 7a-1z7 + 3a-1z9 + 7 - 26z2 + 46z4 - 36z6 + 7z8 + z10 - 10az + 22az3 - 6az5 - 15az7 + 7az9 + 6a2 - 27a2z2 + 51a2z4 - 43a2z6 + 11a2z8 + a2z10 - 10a3z + 28a3z3 - 21a3z5 + 4a3z9 + 2a4 - 7a4z2 + 14a4z4 - 14a4z6 + 7a4z8 - 4a5z + 11a5z3 - 12a5z5 + 7a5z7 + a6z2 - 5a6z4 + 5a6z6 - 3a7z3 + 3a7z5 - a8z2 + a8z4

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

{0, 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 1182. 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                     2   j = 5                   4 1   j = 3                 6 2     j = 1               6 4       j = -1             9 6         j = -3           7 7           j = -5         6 8             j = -7       4 7               j = -9     2 6                 j = -11   1 4                   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, 82]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 82]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 5, 13, 6], X[16, 8, 17, 7], > X[2, 10, 3, 9], X[22, 11, 1, 12], X[18, 13, 19, 14], X[20, 15, 21, 16], > X[6, 18, 7, 17], X[14, 19, 15, 20], X[8, 21, 9, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 82]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -11, 5, -2, 6, -3, 7, -10, 8, -4, 9, -7, 10, > -8, 11, -6]
In[5]:=	DTCode[Knot[11, Alternating, 82]]
Out[5]=	DTCode[4, 10, 12, 16, 2, 22, 18, 20, 6, 14, 8]
In[6]:=	alex = Alexander[Knot[11, Alternating, 82]][t]
Out[6]=	-4 5 12 19 2 3 4 -21 - t + -- - -- + -- + 19 t - 12 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 82]][z]
Out[7]=	4 6 8 1 - 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 116], Knot[11, Alternating, 7], Knot[11, Alternating, 33], > Knot[11, Alternating, 82]}
In[9]:=	{KnotDet[Knot[11, Alternating, 82]], KnotSignature[Knot[11, Alternating, 82]]}
Out[9]=	{95, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 82]][q]
Out[10]=	-7 3 6 10 13 15 15 2 3 4 -12 + q - -- + -- - -- + -- - -- + -- + 10 q - 6 q + 3 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, 33], Knot[11, Alternating, 82]}
In[12]:=	A2Invariant[Knot[11, Alternating, 82]][q]
Out[12]=	-20 -18 2 -14 -12 -10 4 2 2 2 4 6 12 3 + q - q + --- - q - q + q - -- + -- - -- + -- + 3 q - q - q 16 8 6 4 2 q q q q q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 82]][a, z]
Out[13]=	2 4 2 2 4 2 3 z 2 2 4 2 4 z 7 - -- - 6 a + 2 a + 13 z - ---- - 15 a z + 5 a z + 9 z - -- - 2 2 2 a a a 2 4 4 4 6 2 6 4 6 2 8 > 14 a z + 4 a z + 2 z - 6 a z + a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 82]][a, z]
Out[14]=	2 2 2 4 2 z 6 z 3 5 2 8 z 7 + -- + 6 a + 2 a - --- - --- - 10 a z - 10 a z - 4 a z - 26 z - ---- - 2 3 a 2 a a a 3 3 2 2 4 2 6 2 8 2 5 z 13 z 3 3 3 > 27 a z - 7 a z + a z - a z + ---- + ----- + 22 a z + 28 a z + 3 a a 4 5 3 7 3 4 15 z 2 4 4 4 6 4 > 11 a z - 3 a z + 46 z + ----- + 51 a z + 14 a z - 5 a z + 2 a 5 5 8 4 4 z 4 z 5 3 5 5 5 7 5 6 > a z - ---- - ---- - 6 a z - 21 a z - 12 a z + 3 a z - 36 z - 3 a a 6 7 7 12 z 2 6 4 6 6 6 z 7 z 7 5 7 > ----- - 43 a z - 14 a z + 5 a z + -- - ---- - 15 a z + 7 a z + 2 3 a a a 8 9 8 3 z 2 8 4 8 3 z 9 3 9 10 2 10 > 7 z + ---- + 11 a z + 7 a z + ---- + 7 a z + 4 a z + z + a z 2 a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 82]], Vassiliev[3][Knot[11, Alternating, 82]]}
Out[15]=	{0, 2}
In[16]:=	Kh[Knot[11, Alternating, 82]][q, t]
Out[16]=	7 9 1 2 1 4 2 6 4 7 -- + - + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + 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 6 8 7 6 t 2 3 2 3 3 5 3 > ----- + ---- + ---- + --- + 6 q t + 4 q t + 6 q t + 2 q t + 4 q t + 5 2 5 3 q q t q t q t 5 4 7 4 9 5 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a82

K11a81 K11a83