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

K11n57 K11n59

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,5,15,6 X2837 X9,17,10,16 X11,18,12,19 X13,20,14,21 X6,15,7,16 X17,1,18,22 X19,12,20,13 X21,10,22,11

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 -16 -18 -20 6 -22 -12 -10

Alexander Polynomial:

t-3 - 4t-2 + 8t-1 - 9 + 8t - 4t2 + t3

Conway Polynomial:

1 + z2 + 2z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{816, 10156, K11n15, K11n56, ...}

Determinant and Signature:

{35, -2}

Jones Polynomial:

q-5 - 3q-4 + 4q-3 - 5q-2 + 6q-1 - 5 + 5q - 3q2 + 2q3 - q4

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

{K11n56, ...}

A2 (sl(3)) Invariant:

q-16 - q-12 - 2q-8 + q-2 + 3 + q2 + 2q4 - q8 - q12

HOMFLY-PT Polynomial:

- 2a-2 - 3a-2z2 - a-2z4 + 6 + 9z2 + 5z4 + z6 - 4a2 - 6a2z2 - 2a2z4 + a4 + a4z2

Kauffman Polynomial:

- 2a-3z + 7a-3z3 - 5a-3z5 + a-3z7 + 2a-2 - 8a-2z2 + 15a-2z4 - 10a-2z6 + 2a-2z8 - 4a-1z + 10a-1z3 - 5a-1z5 - 2a-1z7 + a-1z9 + 6 - 17z2 + 23z4 - 17z6 + 4z8 - 4az + 6az3 - 5az5 - az7 + az9 + 4a2 - 10a2z2 + 8a2z4 - 6a2z6 + 2a2z8 - 3a3z + 6a3z3 - 5a3z5 + 2a3z7 + a4 + a4z6 - a5z + 3a5z3 + a6z2

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

{1, 1}

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 1158. 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 j = 9                   1 j = 7                 1   j = 5               2 1   j = 3             3 1     j = 1           2 2       j = -1         4 3         j = -3       2 3           j = -5     2 3             j = -7   1 2               j = -9   2                 j = -11 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, NonAlternating, 58]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 58]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 5, 15, 6], X[2, 8, 3, 7], > X[9, 17, 10, 16], X[11, 18, 12, 19], X[13, 20, 14, 21], X[6, 15, 7, 16], > X[17, 1, 18, 22], X[19, 12, 20, 13], X[21, 10, 22, 11]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 58]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -8, 4, -2, -5, 11, -6, 10, -7, -3, 8, 5, -9, 6, -10, > 7, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 58]]
Out[5]=	DTCode[4, 8, 14, 2, -16, -18, -20, 6, -22, -12, -10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 58]][t]
Out[6]=	-3 4 8 2 3 -9 + t - -- + - + 8 t - 4 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 58]][z]
Out[7]=	2 4 6 1 + z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 16], Knot[10, 156], Knot[11, NonAlternating, 15], > Knot[11, NonAlternating, 56], Knot[11, NonAlternating, 58]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 58]], KnotSignature[Knot[11, NonAlternating, 58]]}
Out[9]=	{35, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 58]][q]
Out[10]=	-5 3 4 5 6 2 3 4 -5 + q - -- + -- - -- + - + 5 q - 3 q + 2 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 56], Knot[11, NonAlternating, 58]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 58]][q]
Out[12]=	-16 -12 2 -2 2 4 8 12 3 + q - q - -- + q + q + 2 q - q - q 8 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 58]][a, z]
Out[13]=	2 4 2 2 4 2 3 z 2 2 4 2 4 z 2 4 6 6 - -- - 4 a + a + 9 z - ---- - 6 a z + a z + 5 z - -- - 2 a z + z 2 2 2 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 58]][a, z]
Out[14]=	2 2 2 4 2 z 4 z 3 5 2 8 z 6 + -- + 4 a + a - --- - --- - 4 a z - 3 a z - a z - 17 z - ---- - 2 3 a 2 a a a 3 3 2 2 6 2 7 z 10 z 3 3 3 5 3 4 > 10 a z + a z + ---- + ----- + 6 a z + 6 a z + 3 a z + 23 z + 3 a a 4 5 5 6 15 z 2 4 5 z 5 z 5 3 5 6 10 z > ----- + 8 a z - ---- - ---- - 5 a z - 5 a z - 17 z - ----- - 2 3 a 2 a a a 7 7 8 9 2 6 4 6 z 2 z 7 3 7 8 2 z 2 8 z > 6 a z + a z + -- - ---- - a z + 2 a z + 4 z + ---- + 2 a z + -- + 3 a 2 a a a 9 > a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 58]], Vassiliev[3][Knot[11, NonAlternating, 58]]}
Out[15]=	{1, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 58]][q, t]
Out[16]=	3 4 1 2 1 2 2 3 2 3 t -- + - + ------ + ----- + ----- + ----- + ----- + ---- + ---- + --- + 2 q t + 3 q 11 4 9 3 7 3 7 2 5 2 5 3 q q q t q t q t q t q t q t q t 2 3 2 3 3 5 3 5 4 7 4 9 5 > 2 q t + 3 q t + q t + 2 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n58

K11n57 K11n59