9 47

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9 46.gif

9_46

9 48.gif

9_48

9 47.gif Visit 9 47's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 9 47's page at Knotilus!

Visit 9 47's page at the original Knot Atlas!

9 47 Quick Notes




Simple square depiction
Threefold symmetrical depiction
Ornate threefold symmetrical depiction

Knot presentations

Planar diagram presentation X6271 X16,8,17,7 X8394 X2,15,3,16 X14,9,15,10 X10,6,11,5 X4,14,5,13 X11,1,12,18 X17,13,18,12
Gauss code 1, -4, 3, -7, 6, -1, 2, -3, 5, -6, -8, 9, 7, -5, 4, -2, -9, 8
Dowker-Thistlethwaite code 6 8 10 16 14 -18 4 2 -12
Conway Notation [8*-20]

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index [math]\displaystyle{ \{4,6\} }[/math]
Nakanishi index 2
Maximal Thurston-Bennequin number [-2][-7]
Hyperbolic Volume 10.05
A-Polynomial See Data:9 47/A-polynomial

[edit Notes for 9 47's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 1 }[/math]
Topological 4 genus [math]\displaystyle{ 1 }[/math]
Concordance genus [math]\displaystyle{ 3 }[/math]
Rasmussen s-Invariant -2

[edit Notes for 9 47's four dimensional invariants]

Polynomial invariants

Alexander polynomial [math]\displaystyle{ t^3-4 t^2+6 t-5+6 t^{-1} -4 t^{-2} + t^{-3} }[/math]
Conway polynomial [math]\displaystyle{ z^6+2 z^4-z^2+1 }[/math]
2nd Alexander ideal (db, data sources) [math]\displaystyle{ \{3,t+1\} }[/math]
Determinant and Signature { 27, 2 }
Jones polynomial [math]\displaystyle{ 2 q^5-4 q^4+4 q^3-5 q^2+5 q-3+3 q^{-1} - q^{-2} }[/math]
HOMFLY-PT polynomial (db, data sources) [math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -z^4+4 z^2 a^{-2} -3 z^2 a^{-4} -2 z^2+ a^{-2} -2 a^{-4} + a^{-6} +1 }[/math]
Kauffman polynomial (db, data sources) [math]\displaystyle{ 2 z^7 a^{-1} +2 z^7 a^{-3} +6 z^6 a^{-2} +3 z^6 a^{-4} +3 z^6+a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} -16 z^4 a^{-2} -7 z^4 a^{-4} -9 z^4-2 a z^3+z^3 a^{-1} +6 z^3 a^{-3} +3 z^3 a^{-5} +11 z^2 a^{-2} +9 z^2 a^{-4} +3 z^2 a^{-6} +5 z^2-2 z a^{-1} -5 z a^{-3} -3 z a^{-5} - a^{-2} -2 a^{-4} - a^{-6} +1 }[/math]
The A2 invariant [math]\displaystyle{ -q^6+q^4+q^2+2+2 q^{-2} - q^{-4} + q^{-6} -2 q^{-8} - q^{-12} - q^{-14} + q^{-16} + q^{-20} }[/math]
The G2 invariant [math]\displaystyle{ q^{32}-2 q^{30}+4 q^{28}-7 q^{26}+4 q^{24}-q^{22}-8 q^{20}+16 q^{18}-16 q^{16}+16 q^{14}-4 q^{12}-11 q^{10}+20 q^8-18 q^6+14 q^4-2 q^2-13+21 q^{-2} -8 q^{-4} +13 q^{-8} -19 q^{-10} +18 q^{-12} -4 q^{-14} -9 q^{-16} +12 q^{-18} -21 q^{-20} +25 q^{-22} -13 q^{-24} +9 q^{-28} -19 q^{-30} +23 q^{-32} -18 q^{-34} +4 q^{-36} +3 q^{-38} -14 q^{-40} +19 q^{-42} -11 q^{-44} -3 q^{-46} +15 q^{-48} -19 q^{-50} +12 q^{-52} + q^{-54} -17 q^{-56} +20 q^{-58} -17 q^{-60} +11 q^{-62} +2 q^{-64} -11 q^{-66} +14 q^{-68} -12 q^{-70} +8 q^{-72} -5 q^{-76} +2 q^{-78} -2 q^{-80} +2 q^{-82} - q^{-84} +2 q^{-86} + q^{-88} }[/math]
Further Quantum Invariants
Further quantum knot invariants for 9_47.

A1 Invariants.

Weight Invariant
1 [math]\displaystyle{ -q^5+2 q^3+2 q^{-1} - q^{-5} -2 q^{-9} +2 q^{-11} }[/math]
2 [math]\displaystyle{ q^{16}-2 q^{14}-3 q^{12}+5 q^{10}+2 q^8-5 q^6+3 q^4+6 q^2-4- q^{-2} +4 q^{-4} -3 q^{-6} -4 q^{-8} +2 q^{-10} +2 q^{-12} -3 q^{-14} - q^{-16} +7 q^{-18} - q^{-20} -5 q^{-22} +6 q^{-24} -5 q^{-28} + q^{-30} + q^{-32} }[/math]
3 [math]\displaystyle{ -q^{33}+2 q^{31}+3 q^{29}-2 q^{27}-8 q^{25}-5 q^{23}+11 q^{21}+11 q^{19}-5 q^{17}-17 q^{15}-2 q^{13}+20 q^{11}+15 q^9-14 q^7-20 q^5+6 q^3+24 q- q^{-1} -24 q^{-3} -7 q^{-5} +19 q^{-7} +10 q^{-9} -18 q^{-11} -8 q^{-13} +13 q^{-15} +12 q^{-17} -9 q^{-19} -10 q^{-21} +4 q^{-23} +16 q^{-25} +3 q^{-27} -16 q^{-29} -13 q^{-31} +13 q^{-33} +21 q^{-35} -10 q^{-37} -27 q^{-39} + q^{-41} +27 q^{-43} +5 q^{-45} -17 q^{-47} -12 q^{-49} +11 q^{-51} +10 q^{-53} -3 q^{-55} -4 q^{-57} -2 q^{-59} +2 q^{-61} }[/math]
4 [math]\displaystyle{ q^{56}-2 q^{54}-3 q^{52}+2 q^{50}+5 q^{48}+11 q^{46}-2 q^{44}-16 q^{42}-18 q^{40}-10 q^{38}+29 q^{36}+33 q^{34}+10 q^{32}-26 q^{30}-58 q^{28}-16 q^{26}+35 q^{24}+70 q^{22}+43 q^{20}-51 q^{18}-83 q^{16}-45 q^{14}+56 q^{12}+109 q^{10}+32 q^8-71 q^6-116 q^4-24 q^2+98+97 q^{-2} -5 q^{-4} -109 q^{-6} -72 q^{-8} +49 q^{-10} +97 q^{-12} +32 q^{-14} -69 q^{-16} -68 q^{-18} +17 q^{-20} +70 q^{-22} +30 q^{-24} -42 q^{-26} -53 q^{-28} +54 q^{-32} +32 q^{-34} -20 q^{-36} -52 q^{-38} -34 q^{-40} +32 q^{-42} +60 q^{-44} +38 q^{-46} -47 q^{-48} -95 q^{-50} -23 q^{-52} +75 q^{-54} +116 q^{-56} +9 q^{-58} -120 q^{-60} -102 q^{-62} +18 q^{-64} +135 q^{-66} +84 q^{-68} -58 q^{-70} -108 q^{-72} -55 q^{-74} +60 q^{-76} +84 q^{-78} +16 q^{-80} -38 q^{-82} -49 q^{-84} -6 q^{-86} +22 q^{-88} +17 q^{-90} +4 q^{-92} -8 q^{-94} -5 q^{-96} + q^{-98} + q^{-100} }[/math]
5 [math]\displaystyle{ -q^{85}+2 q^{83}+3 q^{81}-2 q^{79}-5 q^{77}-8 q^{75}-4 q^{73}+7 q^{71}+24 q^{69}+24 q^{67}+3 q^{65}-27 q^{63}-51 q^{61}-48 q^{59}-5 q^{57}+66 q^{55}+95 q^{53}+67 q^{51}-13 q^{49}-112 q^{47}-156 q^{45}-90 q^{43}+58 q^{41}+190 q^{39}+220 q^{37}+93 q^{35}-137 q^{33}-303 q^{31}-264 q^{29}-17 q^{27}+278 q^{25}+409 q^{23}+230 q^{21}-146 q^{19}-452 q^{17}-434 q^{15}-65 q^{13}+380 q^{11}+548 q^9+284 q^7-221 q^5-572 q^3-451 q+40 q^{-1} +495 q^{-3} +539 q^{-5} +130 q^{-7} -371 q^{-9} -539 q^{-11} -236 q^{-13} +254 q^{-15} +480 q^{-17} +282 q^{-19} -146 q^{-21} -402 q^{-23} -276 q^{-25} +86 q^{-27} +317 q^{-29} +229 q^{-31} -48 q^{-33} -254 q^{-35} -192 q^{-37} +48 q^{-39} +206 q^{-41} +151 q^{-43} -34 q^{-45} -183 q^{-47} -159 q^{-49} +13 q^{-51} +173 q^{-53} +191 q^{-55} +55 q^{-57} -150 q^{-59} -255 q^{-61} -160 q^{-63} +92 q^{-65} +319 q^{-67} +312 q^{-69} +19 q^{-71} -345 q^{-73} -469 q^{-75} -191 q^{-77} +296 q^{-79} +590 q^{-81} +398 q^{-83} -168 q^{-85} -624 q^{-87} -572 q^{-89} -30 q^{-91} +534 q^{-93} +663 q^{-95} +242 q^{-97} -364 q^{-99} -619 q^{-101} -370 q^{-103} +129 q^{-105} +472 q^{-107} +404 q^{-109} +42 q^{-111} -276 q^{-113} -319 q^{-115} -135 q^{-117} +99 q^{-119} +199 q^{-121} +136 q^{-123} + q^{-125} -89 q^{-127} -78 q^{-129} -31 q^{-131} +18 q^{-133} +35 q^{-135} +18 q^{-137} -3 q^{-139} -6 q^{-141} -4 q^{-143} -2 q^{-145} +2 q^{-147} }[/math]

A2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ -q^6+q^4+q^2+2+2 q^{-2} - q^{-4} + q^{-6} -2 q^{-8} - q^{-12} - q^{-14} + q^{-16} + q^{-20} }[/math]
1,1 [math]\displaystyle{ q^{20}-4 q^{18}+10 q^{16}-22 q^{14}+32 q^{12}-46 q^{10}+56 q^8-52 q^6+48 q^4-22 q^2+4+36 q^{-2} -58 q^{-4} +74 q^{-6} -94 q^{-8} +90 q^{-10} -94 q^{-12} +70 q^{-14} -52 q^{-16} +28 q^{-18} +5 q^{-20} -24 q^{-22} +50 q^{-24} -56 q^{-26} +58 q^{-28} -46 q^{-30} +32 q^{-32} -22 q^{-34} +10 q^{-36} -2 q^{-38} -4 q^{-40} +2 q^{-46} }[/math]
2,0 [math]\displaystyle{ q^{18}-q^{16}-3 q^{14}-q^{12}+2 q^{10}+2 q^8+4 q^4+6 q^2+3-2 q^{-2} + q^{-4} -3 q^{-6} -3 q^{-8} -2 q^{-10} -2 q^{-12} -2 q^{-14} -2 q^{-16} +3 q^{-18} +3 q^{-24} +5 q^{-26} - q^{-28} - q^{-30} + q^{-32} -2 q^{-38} - q^{-48} + q^{-52} }[/math]

A3 Invariants.

Weight Invariant
0,1,0 [math]\displaystyle{ q^{14}-2 q^{12}+q^8-4 q^6+5 q^4+3 q^2+1+7 q^{-2} +3 q^{-4} -4 q^{-6} -2 q^{-8} -3 q^{-10} -5 q^{-12} -2 q^{-14} +5 q^{-18} + q^{-20} + q^{-22} +5 q^{-24} -4 q^{-26} -3 q^{-28} +3 q^{-30} -3 q^{-32} - q^{-34} +3 q^{-36} }[/math]
1,0,0 [math]\displaystyle{ -q^7+q^5+3 q+ q^{-1} +2 q^{-3} - q^{-11} -2 q^{-15} -2 q^{-19} + q^{-21} + q^{-25} + q^{-27} }[/math]

A4 Invariants.

Weight Invariant
0,1,0,0 [math]\displaystyle{ q^{16}-q^{14}-q^{12}-2 q^8+4 q^4+3 q^2+3+9 q^{-2} +7 q^{-4} + q^{-6} -2 q^{-8} + q^{-10} -6 q^{-12} -12 q^{-14} -5 q^{-16} -6 q^{-20} + q^{-22} +9 q^{-24} +2 q^{-26} +2 q^{-28} +6 q^{-30} +3 q^{-32} -4 q^{-34} - q^{-36} +2 q^{-38} -3 q^{-40} -5 q^{-42} + q^{-44} +2 q^{-46} -2 q^{-48} + q^{-50} +2 q^{-52} }[/math]
1,0,0,0 [math]\displaystyle{ -q^8+q^6+2 q^2+2+ q^{-2} +2 q^{-4} + q^{-8} - q^{-10} + q^{-12} - q^{-14} -2 q^{-18} - q^{-20} - q^{-22} -2 q^{-24} + q^{-26} + q^{-30} + q^{-32} + q^{-34} }[/math]

B2 Invariants.

Weight Invariant
0,1 [math]\displaystyle{ -q^{14}+2 q^{12}-4 q^{10}+5 q^8-4 q^6+5 q^4-3 q^2+3+ q^{-2} - q^{-4} +6 q^{-6} -6 q^{-8} +9 q^{-10} -9 q^{-12} +8 q^{-14} -8 q^{-16} +3 q^{-18} -3 q^{-20} - q^{-22} + q^{-24} -4 q^{-26} +5 q^{-28} -5 q^{-30} +5 q^{-32} -3 q^{-34} +3 q^{-36} }[/math]
1,0 [math]\displaystyle{ q^{24}-2 q^{20}-2 q^{18}+2 q^{16}+3 q^{14}-3 q^{12}-4 q^{10}+q^8+7 q^6+3 q^4-3 q^2-1+6 q^{-2} +5 q^{-4} + q^{-6} -4 q^{-8} - q^{-10} + q^{-12} - q^{-14} -6 q^{-16} -4 q^{-18} + q^{-20} + q^{-22} -2 q^{-24} -4 q^{-26} +3 q^{-28} +6 q^{-30} +2 q^{-32} -3 q^{-34} + q^{-36} +5 q^{-38} +3 q^{-40} -4 q^{-42} -5 q^{-44} + q^{-46} +4 q^{-48} - q^{-50} -4 q^{-52} -2 q^{-54} + q^{-56} +3 q^{-58} }[/math]

D4 Invariants.

Weight Invariant
1,0,0,0 [math]\displaystyle{ q^{18}-2 q^{16}+2 q^{14}-4 q^{12}+4 q^{10}-4 q^8+4 q^6-2 q^4+6 q^2+2+3 q^{-2} +4 q^{-4} +3 q^{-8} -6 q^{-10} +4 q^{-12} -9 q^{-14} +4 q^{-16} -9 q^{-18} +5 q^{-20} -5 q^{-22} +6 q^{-24} -2 q^{-26} +2 q^{-28} + q^{-30} +2 q^{-34} -4 q^{-36} +2 q^{-38} -5 q^{-40} +5 q^{-42} -3 q^{-44} +2 q^{-46} -2 q^{-48} +3 q^{-50} }[/math]

G2 Invariants.

Weight Invariant
1,0 [math]\displaystyle{ q^{32}-2 q^{30}+4 q^{28}-7 q^{26}+4 q^{24}-q^{22}-8 q^{20}+16 q^{18}-16 q^{16}+16 q^{14}-4 q^{12}-11 q^{10}+20 q^8-18 q^6+14 q^4-2 q^2-13+21 q^{-2} -8 q^{-4} +13 q^{-8} -19 q^{-10} +18 q^{-12} -4 q^{-14} -9 q^{-16} +12 q^{-18} -21 q^{-20} +25 q^{-22} -13 q^{-24} +9 q^{-28} -19 q^{-30} +23 q^{-32} -18 q^{-34} +4 q^{-36} +3 q^{-38} -14 q^{-40} +19 q^{-42} -11 q^{-44} -3 q^{-46} +15 q^{-48} -19 q^{-50} +12 q^{-52} + q^{-54} -17 q^{-56} +20 q^{-58} -17 q^{-60} +11 q^{-62} +2 q^{-64} -11 q^{-66} +14 q^{-68} -12 q^{-70} +8 q^{-72} -5 q^{-76} +2 q^{-78} -2 q^{-80} +2 q^{-82} - q^{-84} +2 q^{-86} + q^{-88} }[/math]

.

Computer Talk
The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`
Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:= K = Knot["9 47"];
In[4]:= Alexander[K][t]
KnotTheory::loading: Loading precomputed data in PD4Knots`.
Out[4]= [math]\displaystyle{ t^3-4 t^2+6 t-5+6 t^{-1} -4 t^{-2} + t^{-3} }[/math]
In[5]:= Conway[K][z]
Out[5]= [math]\displaystyle{ z^6+2 z^4-z^2+1 }[/math]
In[6]:= Alexander[K, 2][t]
KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.
Out[6]= [math]\displaystyle{ \{3,t+1\} }[/math]
In[7]:= {KnotDet[K], KnotSignature[K]}
Out[7]= { 27, 2 }
In[8]:= Jones[K][q]
KnotTheory::loading: Loading precomputed data in Jones4Knots`.
Out[8]= [math]\displaystyle{ 2 q^5-4 q^4+4 q^3-5 q^2+5 q-3+3 q^{-1} - q^{-2} }[/math]
In[9]:= HOMFLYPT[K][a, z]
KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.
Out[9]= [math]\displaystyle{ z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -z^4+4 z^2 a^{-2} -3 z^2 a^{-4} -2 z^2+ a^{-2} -2 a^{-4} + a^{-6} +1 }[/math]
In[10]:= Kauffman[K][a, z]
KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.
Out[10]= [math]\displaystyle{ 2 z^7 a^{-1} +2 z^7 a^{-3} +6 z^6 a^{-2} +3 z^6 a^{-4} +3 z^6+a z^5-4 z^5 a^{-1} -4 z^5 a^{-3} +z^5 a^{-5} -16 z^4 a^{-2} -7 z^4 a^{-4} -9 z^4-2 a z^3+z^3 a^{-1} +6 z^3 a^{-3} +3 z^3 a^{-5} +11 z^2 a^{-2} +9 z^2 a^{-4} +3 z^2 a^{-6} +5 z^2-2 z a^{-1} -5 z a^{-3} -3 z a^{-5} - a^{-2} -2 a^{-4} - a^{-6} +1 }[/math]

Vassiliev invariants

V2 and V3: (-1, -2)
V2,1 through V6,9:
V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9
[math]\displaystyle{ -4 }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ 8 }[/math] [math]\displaystyle{ -\frac{110}{3} }[/math] [math]\displaystyle{ -\frac{34}{3} }[/math] [math]\displaystyle{ 64 }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ \frac{32}{3} }[/math] [math]\displaystyle{ -16 }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ 128 }[/math] [math]\displaystyle{ \frac{440}{3} }[/math] [math]\displaystyle{ \frac{136}{3} }[/math] [math]\displaystyle{ \frac{10529}{30} }[/math] [math]\displaystyle{ -\frac{418}{15} }[/math] [math]\displaystyle{ \frac{10738}{45} }[/math] [math]\displaystyle{ -\frac{833}{18} }[/math] [math]\displaystyle{ \frac{1409}{30} }[/math]

V2,1 through V6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V2 and V3.

Khovanov Homology

The coefficients of the monomials [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]2 is the signature of 9 47. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\ r
  \  
j \
 -3-2-101234χ
11       22
9      2 -2
7     22 0
5    32  -1
3   22   0
1  24    2
-1 11     0
-3 2      2
-51       -1

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math] 

In[1]:=    
 
<< KnotTheory`
 
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=
Crossings[Knot[9, 47]]
Out[2]=  
9
In[3]:=
PD[Knot[9, 47]]
Out[3]=  
PD[X[6, 2, 7, 1], X[16, 8, 17, 7], X[8, 3, 9, 4], X[2, 15, 3, 16], 

 X[14, 9, 15, 10], X[10, 6, 11, 5], X[4, 14, 5, 13], X[11, 1, 12, 18], 



  X[17, 13, 18, 12]]
In[4]:=
GaussCode[Knot[9, 47]]
Out[4]=  
GaussCode[1, -4, 3, -7, 6, -1, 2, -3, 5, -6, -8, 9, 7, -5, 4, -2, -9, 8]
In[5]:=
BR[Knot[9, 47]]
Out[5]=  
BR[4, {-1, 2, -1, 2, 3, 2, -1, 2, 3}]
In[6]:=
alex = Alexander[Knot[9, 47]][t]
Out[6]=  
      -3   4    6            2    3

-5 + t   - -- + - + 6 t - 4 t  + t



           2   t


           t
In[7]:=
Conway[Knot[9, 47]][z]
Out[7]=  
     2      4    6
1 - z  + 2 z  + z
In[8]:=
Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=  
{Knot[9, 47]}
In[9]:=
{KnotDet[Knot[9, 47]], KnotSignature[Knot[9, 47]]}
Out[9]=  
{27, 2}
In[10]:=
J=Jones[Knot[9, 47]][q]
Out[10]=  
      -2   3            2      3      4      5

-3 - q   + - + 5 q - 5 q  + 4 q  - 4 q  + 2 q


           q
In[11]:=
Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=  
{Knot[9, 47]}
In[12]:=
A2Invariant[Knot[9, 47]][q]
Out[12]=  
     -6    -4    -2      2    4    6      8    12    14    16    20
2 - q   + q   + q   + 2 q  - q  + q  - 2 q  - q   - q   + q   + q
In[13]:=
Kauffman[Knot[9, 47]][a, z]
Out[13]=  
                                                 2      2       2

    -6   2     -2   3 z   5 z   2 z      2   3 z    9 z    11 z



1 - a   - -- - a   - --- - --- - --- + 5 z  + ---- + ---- + ----- + 



          4          5     3     a             6      4      2
         a          a     a                   a      a      a

    3      3    3                      4       4    5      5      5
 3 z    6 z    z         3      4   7 z    16 z    z    4 z    4 z
 ---- + ---- + -- - 2 a z  - 9 z  - ---- - ----- + -- - ---- - ---- + 
   5      3    a                      4      2      5     3     a
  a      a                           a      a      a     a

                  6      6      7      7
    5      6   3 z    6 z    2 z    2 z
 a z  + 3 z  + ---- + ---- + ---- + ----
                 4      2      3     a


                 a      a      a
In[14]:=
{Vassiliev[2][Knot[9, 47]], Vassiliev[3][Knot[9, 47]]}
Out[14]=  
{0, -2}
In[15]:=
Kh[Knot[9, 47]][q, t]
Out[15]=  
         3     1       2      1      1    2 q      3        5

4 q + 2 q  + ----- + ----- + ---- + --- + --- + 2 q  t + 3 q  t + 



             5  3    3  2      2   q t    t
            q  t    q  t    q t

    5  2      7  2      7  3      9  3      11  4


  2 q  t  + 2 q  t  + 2 q  t  + 2 q  t  + 2 q   t
Retrieved from "https://katlas.org/index.php?title=9_47&oldid=25796"