The Alternating Knot 10₉₈

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Acknowledgement

KnotPlot

PD Presentation: X₁₆₂₇ X_3,10,4,11 X_7,18,8,19 X_17,8,18,9 X_9,2,10,3 X_11,16,12,17 X_5,15,6,14 X_15,5,16,4 X_13,20,14,1 X_19,12,20,13

Gauss Code: {-1, 5, -2, 8, -7, 1, -3, 4, -5, 2, -6, 10, -9, 7, -8, 6, -4, 3, -10, 9}

DT (Dowker-Thistlethwaite) Code: 6 10 14 18 2 16 20 4 8 12

Minimum Braid Representative:

Length is 11, width is 4
Braid index is 4

A Morse Link Presentation:

3D Invariants:

Symmetry Type Unknotting Number 3-Genus Bridge/Super Bridge Index Nakanishi Index

Chiral 2 3 3 / NotAvailable 2

Alexander Polynomial: - 2t^-3 + 9t^-2 - 18t^-1 + 23 - 18t + 9t² - 2t³

Conway Polynomial: 1 - 3z⁴ - 2z⁶

Other knots with the same Alexander/Conway Polynomial: {10₈₇, K11a58, K11a165, K11n72, ...}

Determinant and Signature: {81, -4}

Jones Polynomial: q^-10 - 3q^-9 + 7q^-8 - 11q^-7 + 12q^-6 - 14q^-5 + 13q^-4 - 9q^-3 + 7q^-2 - 3q^-1 + 1

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

A2 (sl(3)) Invariant: q^-30 - q^-28 + 2q^-26 + 2q^-24 - 3q^-22 - 5q^-18 - q^-16 + q^-14 + 5q^-10 - q^-8 + 2q^-6 + q^-4 - q^-2 + 1

HOMFLY-PT Polynomial: a² + 2a²z² + a²z⁴ + 3a⁴ + a⁴z² - 2a⁴z⁴ - a⁴z⁶ - 5a⁶ - 5a⁶z² - 3a⁶z⁴ - a⁶z⁶ + 2a⁸ + 2a⁸z² + a⁸z⁴

Kauffman Polynomial: - a² + 3a²z² - 3a²z⁴ + a²z⁶ + 5a³z³ - 8a³z⁵ + 3a³z⁷ + 3a⁴ - 2a⁴z² + 4a⁴z⁴ - 9a⁴z⁶ + 4a⁴z⁸ - 6a⁵z + 14a⁵z³ - 17a⁵z⁵ + 3a⁵z⁷ + 2a⁵z⁹ + 5a⁶ - 10a⁶z² + 17a⁶z⁴ - 23a⁶z⁶ + 10a⁶z⁸ - 12a⁷z + 25a⁷z³ - 26a⁷z⁵ + 8a⁷z⁷ + 2a⁷z⁹ + 2a⁸ + 2a⁸z⁴ - 7a⁸z⁶ + 6a⁸z⁸ - 6a⁹z + 14a⁹z³ - 14a⁹z⁵ + 8a⁹z⁷ + 4a¹⁰z² - 7a¹⁰z⁴ + 6a¹⁰z⁶ - 2a¹¹z³ + 3a¹¹z⁵ - a¹²z² + a¹²z⁴

V₂ and V₃, the type 2 and 3 Vassiliev invariants: {0, 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=-4 is the signature of 10₉₈. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

t^rq^j r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2

j = 1 1

j = -1 2

j = -3 5 1

j = -5 5 3

j = -7 8 4

j = -9 6 5

j = -11 6 8

j = -13 5 6

j = -15 2 6

j = -17 1 5

j = -19 2

j = -21 1

n Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2 q^-28 - 3q^-27 + 3q^-26 + 5q^-25 - 19q^-24 + 16q^-23 + 22q^-22 - 62q^-21 + 33q^-20 + 63q^-19 - 116q^-18 + 37q^-17 + 110q^-16 - 147q^-15 + 17q^-14 + 136q^-13 - 137q^-12 - 13q^-11 + 130q^-10 - 93q^-9 - 33q^-8 + 94q^-7 - 40q^-6 - 34q^-5 + 45q^-4 - 7q^-3 - 17q^-2 + 11q^-1 + 1 - 3q + q²

3 q^-54 - 3q^-53 + 3q^-52 + q^-51 - 3q^-50 - 6q^-49 + 13q^-48 + 9q^-47 - 31q^-46 - 21q^-45 + 70q^-44 + 42q^-43 - 118q^-42 - 104q^-41 + 193q^-40 + 195q^-39 - 261q^-38 - 325q^-37 + 304q^-36 + 495q^-35 - 329q^-34 - 649q^-33 + 286q^-32 + 807q^-31 - 238q^-30 - 894q^-29 + 131q^-28 + 961q^-27 - 34q^-26 - 955q^-25 - 83q^-24 + 920q^-23 + 193q^-22 - 844q^-21 - 285q^-20 + 718q^-19 + 383q^-18 - 599q^-17 - 418q^-16 + 421q^-15 + 450q^-14 - 279q^-13 - 403q^-12 + 121q^-11 + 349q^-10 - 26q^-9 - 245q^-8 - 47q^-7 + 163q^-6 + 58q^-5 - 80q^-4 - 55q^-3 + 35q^-2 + 34q^-1 - 10 - 17q + 3q² + 5q³ + q⁴ - 3q⁵ + q⁶

4 q^-88 - 3q^-87 + 3q^-86 + q^-85 - 7q^-84 + 10q^-83 - 9q^-82 + 10q^-81 - 2q^-80 - 36q^-79 + 47q^-78 - 3q^-77 + 36q^-76 - 38q^-75 - 170q^-74 + 121q^-73 + 121q^-72 + 216q^-71 - 147q^-70 - 664q^-69 + 48q^-68 + 474q^-67 + 964q^-66 - 46q^-65 - 1754q^-64 - 758q^-63 + 710q^-62 + 2591q^-61 + 978q^-60 - 2975q^-59 - 2607q^-58 - 5q^-57 + 4448q^-56 + 3163q^-55 - 3280q^-54 - 4700q^-53 - 1880q^-52 + 5354q^-51 + 5546q^-50 - 2340q^-49 - 5826q^-48 - 4023q^-47 + 4939q^-46 + 6975q^-45 - 828q^-44 - 5652q^-43 - 5511q^-42 + 3709q^-41 + 7197q^-40 + 638q^-39 - 4600q^-38 - 6208q^-37 + 2109q^-36 + 6557q^-35 + 1950q^-34 - 3014q^-33 - 6245q^-32 + 266q^-31 + 5172q^-30 + 3002q^-29 - 997q^-28 - 5467q^-27 - 1488q^-26 + 3057q^-25 + 3274q^-24 + 1000q^-23 - 3698q^-22 - 2366q^-21 + 746q^-20 + 2368q^-19 + 2073q^-18 - 1505q^-17 - 1910q^-16 - 712q^-15 + 855q^-14 + 1774q^-13 - 20q^-12 - 770q^-11 - 853q^-10 - 139q^-9 + 815q^-8 + 313q^-7 - 17q^-6 - 366q^-5 - 280q^-4 + 177q^-3 + 125q^-2 + 109q^-1 - 57 - 109q + 15q² + 7q³ + 36q⁴ + 2q⁵ - 20q⁶ + 3q⁷ - 3q⁸ + 5q⁹ + q¹⁰ - 3q¹¹ + q¹²

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]:=
PD[Knot[10, 98]]

Out[2]=
PD[X[1, 6, 2, 7], X[3, 10, 4, 11], X[7, 18, 8, 19], X[17, 8, 18, 9], > X[9, 2, 10, 3], X[11, 16, 12, 17], X[5, 15, 6, 14], X[15, 5, 16, 4], > X[13, 20, 14, 1], X[19, 12, 20, 13]]

In[3]:=
GaussCode[Knot[10, 98]]

Out[3]=
GaussCode[-1, 5, -2, 8, -7, 1, -3, 4, -5, 2, -6, 10, -9, 7, -8, 6, -4, 3, -10, > 9]

In[4]:=
DTCode[Knot[10, 98]]

Out[4]=
DTCode[6, 10, 14, 18, 2, 16, 20, 4, 8, 12]

In[5]:=
br = BR[Knot[10, 98]]

Out[5]=
BR[4, {-1, -1, -2, -2, 3, -2, 1, -2, -2, 3, -2}]

In[6]:=
{First[br], Crossings[br]}

Out[6]=
{4, 11}

In[7]:=
BraidIndex[Knot[10, 98]]

Out[7]=
4

In[8]:=
Show[DrawMorseLink[Knot[10, 98]]]

Out[8]=
-Graphics-

In[9]:=
#[Knot[10, 98]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}

Out[9]=
{Chiral, 2, 3, 3, NotAvailable, 2}

In[10]:=
alex = Alexander[Knot[10, 98]][t]

Out[10]=
2 9 18 2 3 23 - -- + -- - -- - 18 t + 9 t - 2 t 3 2 t t t

In[11]:=
Conway[Knot[10, 98]][z]

Out[11]=
4 6 1 - 3 z - 2 z

In[12]:=
Select[AllKnots[], (alex === Alexander[#][t])&]

Out[12]=
{Knot[10, 87], Knot[10, 98], Knot[11, Alternating, 58], > Knot[11, Alternating, 165], Knot[11, NonAlternating, 72]}

In[13]:=
{KnotDet[Knot[10, 98]], KnotSignature[Knot[10, 98]]}

Out[13]=
{81, -4}

In[14]:=
Jones[Knot[10, 98]][q]

Out[14]=
-10 3 7 11 12 14 13 9 7 3 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q

In[15]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]

Out[15]=
{Knot[10, 98]}

In[16]:=
A2Invariant[Knot[10, 98]][q]

Out[16]=
-30 -28 2 2 3 5 -16 -14 5 -8 2 -4 1 + q - q + --- + --- - --- - --- - q + q + --- - q + -- + q - 26 24 22 18 10 6 q q q q q q -2 > q

In[17]:=
HOMFLYPT[Knot[10, 98]][a, z]

Out[17]=
2 4 6 8 2 2 4 2 6 2 8 2 2 4 a + 3 a - 5 a + 2 a + 2 a z + a z - 5 a z + 2 a z + a z - 4 4 6 4 8 4 4 6 6 6 > 2 a z - 3 a z + a z - a z - a z

In[18]:=
Kauffman[Knot[10, 98]][a, z]

Out[18]=
2 4 6 8 5 7 9 2 2 4 2 -a + 3 a + 5 a + 2 a - 6 a z - 12 a z - 6 a z + 3 a z - 2 a z - 6 2 10 2 12 2 3 3 5 3 7 3 9 3 > 10 a z + 4 a z - a z + 5 a z + 14 a z + 25 a z + 14 a z - 11 3 2 4 4 4 6 4 8 4 10 4 12 4 > 2 a z - 3 a z + 4 a z + 17 a z + 2 a z - 7 a z + a z - 3 5 5 5 7 5 9 5 11 5 2 6 4 6 > 8 a z - 17 a z - 26 a z - 14 a z + 3 a z + a z - 9 a z - 6 6 8 6 10 6 3 7 5 7 7 7 9 7 > 23 a z - 7 a z + 6 a z + 3 a z + 3 a z + 8 a z + 8 a z + 4 8 6 8 8 8 5 9 7 9 > 4 a z + 10 a z + 6 a z + 2 a z + 2 a z

In[19]:=
{Vassiliev[2][Knot[10, 98]], Vassiliev[3][Knot[10, 98]]}

Out[19]=
{0, 3}

In[20]:=
Kh[Knot[10, 98]][q, t]

Out[20]=
3 5 1 2 1 5 2 6 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 6 6 8 6 5 8 4 5 t 2 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + --- + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q t q 2 > q t

In[21]:=
ColouredJones[Knot[10, 98], 2][q]

Out[21]=
-28 3 3 5 19 16 22 62 33 63 116 37 1 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + 27 26 25 24 23 22 21 20 19 18 17 q q q q q q q q q q q 110 147 17 136 137 13 130 93 33 94 40 34 45 > --- - --- + --- + --- - --- - --- + --- - -- - -- + -- - -- - -- + -- - 16 15 14 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q q q q 7 17 11 2 > -- - -- + -- - 3 q + q 3 2 q q q

Dror Bar-Natan: The Knot Atlas: The Rolfsen Knot Table: The Knot 10₉₈

10₉₇
10₉₉

n	Coloured Jones Polynomial (in the (n+1)-dimensional representation of sl(2))
2	q^-28 - 3q^-27 + 3q^-26 + 5q^-25 - 19q^-24 + 16q^-23 + 22q^-22 - 62q^-21 + 33q^-20 + 63q^-19 - 116q^-18 + 37q^-17 + 110q^-16 - 147q^-15 + 17q^-14 + 136q^-13 - 137q^-12 - 13q^-11 + 130q^-10 - 93q^-9 - 33q^-8 + 94q^-7 - 40q^-6 - 34q^-5 + 45q^-4 - 7q^-3 - 17q^-2 + 11q^-1 + 1 - 3q + q²
3	q^-54 - 3q^-53 + 3q^-52 + q^-51 - 3q^-50 - 6q^-49 + 13q^-48 + 9q^-47 - 31q^-46 - 21q^-45 + 70q^-44 + 42q^-43 - 118q^-42 - 104q^-41 + 193q^-40 + 195q^-39 - 261q^-38 - 325q^-37 + 304q^-36 + 495q^-35 - 329q^-34 - 649q^-33 + 286q^-32 + 807q^-31 - 238q^-30 - 894q^-29 + 131q^-28 + 961q^-27 - 34q^-26 - 955q^-25 - 83q^-24 + 920q^-23 + 193q^-22 - 844q^-21 - 285q^-20 + 718q^-19 + 383q^-18 - 599q^-17 - 418q^-16 + 421q^-15 + 450q^-14 - 279q^-13 - 403q^-12 + 121q^-11 + 349q^-10 - 26q^-9 - 245q^-8 - 47q^-7 + 163q^-6 + 58q^-5 - 80q^-4 - 55q^-3 + 35q^-2 + 34q^-1 - 10 - 17q + 3q² + 5q³ + q⁴ - 3q⁵ + q⁶
4	q^-88 - 3q^-87 + 3q^-86 + q^-85 - 7q^-84 + 10q^-83 - 9q^-82 + 10q^-81 - 2q^-80 - 36q^-79 + 47q^-78 - 3q^-77 + 36q^-76 - 38q^-75 - 170q^-74 + 121q^-73 + 121q^-72 + 216q^-71 - 147q^-70 - 664q^-69 + 48q^-68 + 474q^-67 + 964q^-66 - 46q^-65 - 1754q^-64 - 758q^-63 + 710q^-62 + 2591q^-61 + 978q^-60 - 2975q^-59 - 2607q^-58 - 5q^-57 + 4448q^-56 + 3163q^-55 - 3280q^-54 - 4700q^-53 - 1880q^-52 + 5354q^-51 + 5546q^-50 - 2340q^-49 - 5826q^-48 - 4023q^-47 + 4939q^-46 + 6975q^-45 - 828q^-44 - 5652q^-43 - 5511q^-42 + 3709q^-41 + 7197q^-40 + 638q^-39 - 4600q^-38 - 6208q^-37 + 2109q^-36 + 6557q^-35 + 1950q^-34 - 3014q^-33 - 6245q^-32 + 266q^-31 + 5172q^-30 + 3002q^-29 - 997q^-28 - 5467q^-27 - 1488q^-26 + 3057q^-25 + 3274q^-24 + 1000q^-23 - 3698q^-22 - 2366q^-21 + 746q^-20 + 2368q^-19 + 2073q^-18 - 1505q^-17 - 1910q^-16 - 712q^-15 + 855q^-14 + 1774q^-13 - 20q^-12 - 770q^-11 - 853q^-10 - 139q^-9 + 815q^-8 + 313q^-7 - 17q^-6 - 366q^-5 - 280q^-4 + 177q^-3 + 125q^-2 + 109q^-1 - 57 - 109q + 15q² + 7q³ + 36q⁴ + 2q⁵ - 20q⁶ + 3q⁷ - 3q⁸ + 5q⁹ + q¹⁰ - 3q¹¹ + q¹²

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	PD[Knot[10, 98]]
Out[2]=	PD[X[1, 6, 2, 7], X[3, 10, 4, 11], X[7, 18, 8, 19], X[17, 8, 18, 9], > X[9, 2, 10, 3], X[11, 16, 12, 17], X[5, 15, 6, 14], X[15, 5, 16, 4], > X[13, 20, 14, 1], X[19, 12, 20, 13]]
In[3]:=	GaussCode[Knot[10, 98]]
Out[3]=	GaussCode[-1, 5, -2, 8, -7, 1, -3, 4, -5, 2, -6, 10, -9, 7, -8, 6, -4, 3, -10, > 9]
In[4]:=	DTCode[Knot[10, 98]]
Out[4]=	DTCode[6, 10, 14, 18, 2, 16, 20, 4, 8, 12]
In[5]:=	br = BR[Knot[10, 98]]
Out[5]=	BR[4, {-1, -1, -2, -2, 3, -2, 1, -2, -2, 3, -2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{4, 11}
In[7]:=	BraidIndex[Knot[10, 98]]
Out[7]=	4
In[8]:=	Show[DrawMorseLink[Knot[10, 98]]]

Out[8]=	-Graphics-
In[9]:=	#[Knot[10, 98]]& /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Chiral, 2, 3, 3, NotAvailable, 2}
In[10]:=	alex = Alexander[Knot[10, 98]][t]
Out[10]=	2 9 18 2 3 23 - -- + -- - -- - 18 t + 9 t - 2 t 3 2 t t t
In[11]:=	Conway[Knot[10, 98]][z]
Out[11]=	4 6 1 - 3 z - 2 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[10, 87], Knot[10, 98], Knot[11, Alternating, 58], > Knot[11, Alternating, 165], Knot[11, NonAlternating, 72]}
In[13]:=	{KnotDet[Knot[10, 98]], KnotSignature[Knot[10, 98]]}
Out[13]=	{81, -4}
In[14]:=	Jones[Knot[10, 98]][q]
Out[14]=	-10 3 7 11 12 14 13 9 7 3 1 + q - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q q q q q q q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[10, 98]}
In[16]:=	A2Invariant[Knot[10, 98]][q]
Out[16]=	-30 -28 2 2 3 5 -16 -14 5 -8 2 -4 1 + q - q + --- + --- - --- - --- - q + q + --- - q + -- + q - 26 24 22 18 10 6 q q q q q q -2 > q
In[17]:=	HOMFLYPT[Knot[10, 98]][a, z]
Out[17]=	2 4 6 8 2 2 4 2 6 2 8 2 2 4 a + 3 a - 5 a + 2 a + 2 a z + a z - 5 a z + 2 a z + a z - 4 4 6 4 8 4 4 6 6 6 > 2 a z - 3 a z + a z - a z - a z
In[18]:=	Kauffman[Knot[10, 98]][a, z]
Out[18]=	2 4 6 8 5 7 9 2 2 4 2 -a + 3 a + 5 a + 2 a - 6 a z - 12 a z - 6 a z + 3 a z - 2 a z - 6 2 10 2 12 2 3 3 5 3 7 3 9 3 > 10 a z + 4 a z - a z + 5 a z + 14 a z + 25 a z + 14 a z - 11 3 2 4 4 4 6 4 8 4 10 4 12 4 > 2 a z - 3 a z + 4 a z + 17 a z + 2 a z - 7 a z + a z - 3 5 5 5 7 5 9 5 11 5 2 6 4 6 > 8 a z - 17 a z - 26 a z - 14 a z + 3 a z + a z - 9 a z - 6 6 8 6 10 6 3 7 5 7 7 7 9 7 > 23 a z - 7 a z + 6 a z + 3 a z + 3 a z + 8 a z + 8 a z + 4 8 6 8 8 8 5 9 7 9 > 4 a z + 10 a z + 6 a z + 2 a z + 2 a z
In[19]:=	{Vassiliev[2][Knot[10, 98]], Vassiliev[3][Knot[10, 98]]}
Out[19]=	{0, 3}
In[20]:=	Kh[Knot[10, 98]][q, t]
Out[20]=	3 5 1 2 1 5 2 6 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 21 8 19 7 17 7 17 6 15 6 15 5 13 5 q q q t q t q t q t q t q t q t 6 6 8 6 5 8 4 5 t 2 t > ------ + ------ + ------ + ----- + ----- + ----- + ---- + ---- + -- + --- + 13 4 11 4 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q t q t q 2 > q t
In[21]:=	ColouredJones[Knot[10, 98], 2][q]
Out[21]=	-28 3 3 5 19 16 22 62 33 63 116 37 1 + q - --- + --- + --- - --- + --- + --- - --- + --- + --- - --- + --- + 27 26 25 24 23 22 21 20 19 18 17 q q q q q q q q q q q 110 147 17 136 137 13 130 93 33 94 40 34 45 > --- - --- + --- + --- - --- - --- + --- - -- - -- + -- - -- - -- + -- - 16 15 14 13 12 11 10 9 8 7 6 5 4 q q q q q q q q q q q q q 7 17 11 2 > -- - -- + -- - 3 q + q 3 2 q q q

The Alternating Knot 1098

The Alternating Knot 10₉₈